Introduction - Yale Universityusers.math.yale.edu/~ib93/limits.pdf · Introduction When Γ is a ﬁnitely generated group, let D(Γ) be the set of all representations ρ: Γ → PSL

ALGEBRAIC AND GEOMETRIC CONVERGENCE OFDISCRETE REPRESENTATIONS INTO PSL2 C

IAN BIRINGER & JUAN SOUTO

Abstract. Anderson and Canary have shown that if the algebraiclimit of a sequence of discrete, faithful representations of a finitelygenerated group into PSL2 C does not contain parabolics, then itis also the sequence’s geometric limit. We construct examples thatdemonstrate the failure of this theorem for certain sequences ofunfaithful representations, and offer a suitable replacement.

1. Introduction

When Γ is a finitely generated group, let D(Γ) be the set of allrepresentations ρ : Γ → PSL2 C with discrete, torsion free and non-elementary image. Here, a discrete subgroup of PSL2 C is called elemen-tary if it is virtually abelian. The set D(Γ) sits naturally in the varietyHom(Γ,PSL2 C) and inherits the topology given by pointwise conver-gence; this is called the algebraic topology on D(Γ) and a pointwiseconvergent sequence of representations is usually called algebraicallyconvergent. The goal of this note is to investigate the relationship be-tween the algebraic convergence of a sequence (ρi) in D(Γ) and thegeometric convergence of the subgroups ρi(Γ) ⊂ PSL2 C. Recall that asequence of closed subgroups (Gi) of PSL2 C converges to a subgroupG ⊂ PSL2 C geometrically if it does in the Chabauty topology.

Assume from now on that (ρi) is a sequence in D(Γ) converging al-gebraically to a representation ρ ∈ D(Γ), and that the groups ρi(Γ)converge geometrically to a subgroup G of PSL2 C. For convenience,we will often say that ρi converges geometrically to G. While it is clearthat ρ(Γ) ⊂ G, it was Jorgensen [Jor73] who first realized that thegeometric limit may be larger than the algebraic limit. For example,Thurston [Thu86B] constructed an algebraically convergent sequence inD(π1(Σg)) that converges geometrically to a subgroup of PSL2 C that isnot even finitely generated. Other examples of this phenomenon, eachone dramatic in its own way, were constructed by Kerckhoff-Thurston[KT90], Anderson-Canary [AC96A] and Brock [Bro01].

The second author has been partially supported by the NSF grant DMS-0706878and the Alfred P. Sloan Foundation.

1

2 IAN BIRINGER & JUAN SOUTO

All these examples are related to the appearance of new parabolicelements in the algebraic limit; in fact, the following holds:

Theorem 1.1 (Anderson-Canary[AC96B]). Let Γ be a finitely gener-ated group and assume that (ρi) is a sequence of faithful representationsin D(Γ) converging algebraically to some ρ ∈ D(Γ). If ρ(Γ) does notcontain parabolic elements, then the groups ρi(Γ) converge geometricallyto ρ(Γ).

In [AC00], Anderson and Canary extended this result to the casewhere ρ and ρi map the same elements to parabolics for all i. Evansproved in [Eva04] that the same conclusion holds under the weakerassumption that if an element of Γ is sent to a parabolic by ρ then itis also parabolic in ρi for all i. All these results were obtained in thepresence of certain technical assumptions rendered unnecessary by workof Brock and Souto [BS06], and later by the resolution of the tamenessconjecture by Agol [Ago] and Calegari-Gabai [CG06].

Motivated by questions of a different nature, related to the attempt tounderstand the structure of closed hyperbolic 3-manifolds whose fun-damental group can be generated by, say, 10 elements, we revisitedTheorem 1.1 convinced that it would remain true after dropping theassumption that the representations ρi are faithful. To our surprise, wefound the following examples showing that Theorem 1.1 fails dramati-cally in this more general setting.

Example 1. Let Γ be the fundamental group of a closed surface of genus3. There is a sequence of representations (ρi) in D(Γ) converging alge-braically to a faithful representation ρ and geometrically to a subgroupG ⊂ PSL2 C, such that

• G does not contain any parabolic elements.• ρ(Γ) has index 2 in G.

Example 2. Let Γ be the fundamental group of a compression bodywith exterior boundary of genus 4 and connected interior boundary ofgenus 3. There is a sequence of representations (ρi) in D(Γ) convergingalgebraically to a faithful representation ρ and geometrically to a groupG, such that

• G does not contain any parabolic elements.• ρ(Γ) has infinite index in G.

Example 3. Let Γ be the fundamental group of a compression bodywith exterior boundary of genus 4 and connected interior boundary ofgenus 3. There is a sequence of representations (ρi) in D(Γ) converging

3

algebraically to a faithful representation ρ and geometrically to a groupG, such that

• ρ(Γ) does not contain any parabolic elements.• G is not finitely generated.

Apart from discussing the examples above, our main goal is to under-stand the failure of the Theorem 1.1 and see what is still true withoutthe assumption that the representations are faithful. We prove:

Theorem 1.2. Let Γ be a finitely generated group and (ρi) a sequencein D(Γ). Assume that (ρi) is algebraically convergent and convergesgeometrically to a subgroup G of PSL2 C. If G does not contain parabolicelements, then G is finitely generated.

Before going further, observe that Example 3 shows that for the con-clusion of Theorem 1.2 to hold it does not suffice to assume that thealgebraic limit has no parabolics. Similarly, Example 2 shows that un-der the assumptions of Theorem 1.2 the algebraic limit ρ(Γ) can haveinfinite index in the geometric limit G.

The reader may find it surprising that we mention Example 1 at all;the other two seem to be much more dramatic. However, Example 1 isthe mother of all examples. It also shows that the following theorem ofAnderson fails if one considers non-faithful representations:

Theorem 1.3 (Anderson). Assume that Γ is a finitely generated group,and that (ρi) is a sequence of faithful representations in D(Γ) convergingalgebraically to some representation ρ and geometrically to a subgroup Gof PSL2 C. Then maximal cyclic subgroups of ρ(Γ) are maximal cyclicin G. In particular, if the image ρ(Γ) of the algebraic limit has finiteindex in the geometric limit G, then ρ(Γ) = G.

We mention Theorem 1.3 because its failure is the heart of the failureof Theorem 1.1 for sequences which are not necessarily faithful:

Theorem 1.4. Let Γ be a finitely generated group and (ρi) a sequencein D(Γ). Assume that (ρi) converges algebraically to a representation ρand geometrically to a subgroup G of PSL2 C. If

• ρ(Γ) does not contain parabolic elements, and• maximal cyclic subgroups of ρ(Γ) are maximal cyclic in G,

then G = ρ(Γ).

The proof of Theorem 1.4, as the proof of all results in this note, iscompletely independent of Theorem 1.1. In fact, combining Theorem1.4 with Anderson’s Theorem 1.3 we obtain a new simpler proof of The-orem 1.1. It should be observed that while Anderson-Canary and Evans


used fairly involved arguments to bypass the question of tameness, weuse here the resolution of the tameness conjecture by Agol [Ago] andCalegari-Gabai [CG06] in a crucial way.

To conclude this fairly long introduction, we describe the paper sec-tion by section. After some preliminaries in section 2 we construct theexamples mentioned above in section 3. In section 4 we prove Theorem1.4, obtaining a new proof of Theorem 1.1. In section 5 we obtain atechnical result, Proposition 5.1, which asserts that under some con-ditions if M → N is a cover then there is a tower M → M ′ → Nwhere χ(M ′) > χ(M). Proposition 5.1 and a slightly stronger versionof Theorem 1.4 are the key ingredients in the proof of Theorem 1.2 insection 6: the idea is to show that if H ⊂ G is finitely generated, con-tains ρ(Γ) and is such that χ(H3/H) is maximal then G = H and henceG is finitely generated. In section 7 we discuss extensions of Theorem1.2 and Theorem 1.4 to the case that the algebraic limit has cusps; forinstance, this permits us to recover Evans’ general version of Theorem1.1. Once this is done, we discuss briefly in appendix A to which extentother well-known theorems about faithful representations remain trueif the condition of faithfulness is dropped. We conclude with a remark,in appendix B, which may be of independent interest. We describea possible strategy to construct a closed hyperbolic 3-manifold whoseHeegaard genus is larger than the rank of its fundamental group; thisstrategy is motived by the examples presented in section 3.

The authors would like to thank Misha Kapovich for the idea of usingdimensions of deformation spaces to prove Proposition 5.1.

2. Preliminaries

In this section we recall some well-known facts and definitions abouthyperbolic 3-manifolds.

2.1. Hyperbolic manifolds. By a hyperbolic 3-manifold we will meanany Riemannian 3-manifold isometric to H3/Γ, where Γ is a discrete,torsion-free group of isometries of hyperbolic 3-space. We will usuallyassume that the elements of Γ are orientation preserving, or equivalentlythat H3/Γ is orientable. The full group of orientation preserving isome-tries of hyperbolic 3-space is written Isom+(H3), and is often identifiedwith PSL2 C through its action on the boundary of H3.

Conjugate subgroups of PSL2 C give isometric quotients of H3; inorder to remove this indeterminacy we consider pointed hyperbolic 3-manifolds, i.e. pairs (M,ω) where ω is an orthonormal frame of sometangent space of M . Choosing once and for ever a fixed frame ωH3 of

5

some tangent space of H3, every quotient manifold H3/Γ has an inducedframing ωH3/Γ given by the projection of ωH3 . Now, if (M,ω) is a pointedhyperbolic 3-manifold then there is a unique Γ ⊂ PSL2 C such that themanifolds (M,ω) and (H3/Γ, ωH3/Γ) are isometric as pointed manifolds.

Remark. It would be more natural to speak of framed hyperbolic 3-manifolds instead of pointed; however, it is customary to use the giventerminology.

2.2. Tameness. We will be mainly interested in hyperbolic 3-manifoldsM with finitely generated fundamental group. Any such manifold istame by the work of Agol [Ago] and Calegari-Gabai [CG06]:

Tameness Theorem (Agol, Calegari-Gabai). Let M be a hyperbolic3-manifold with finitely generated fundamental group. Then M is home-omorphic to the interior of a compact 3-manifold.

A standard core of a hyperbolic 3-manifold M with finitely generatedfundamental group is a compact submanifold C ⊂M with M \C home-omorphic to ∂C ×R. Observe that M is homeomorphic to the interiorof every such compact core. It follows from the Tameness Theorem thatevery such M admits an exhaustion by nested compact cores. If C ⊂Mis a standard compact core, then the ends of M correspond naturallyto components of M \ C. The component UE of M \ C correspondingto an end E is said to be a standard neighborhood of E and the com-ponent of ∂C contained in the closure of UE is said to face E . We willoften denote the component of ∂C facing E by ∂E . Observe that UE ishomeomorphic to ∂E × R.

2.3. Geometry of ends in the absence of cusps. LetM be a hyper-bolic 3-manifold with finitely generated fundamental group and withoutcusps. An end E of M is convex cocompact if it has a neighborhood inM disjoint from the convex-core CC(M) of M . Recall that the convexcore CC(M) is the smallest convex submanifold of M whose inclusionis a homotopy equivalence. A manifold with compact CC(M) is said tobe convex cocompact; equivalently, all ends of M are convex cocompact.

For every d > 0, the set of points in M within distance d of CC(M) ishomeomorphic to M and has strictly convex C1-boundary. Smoothingthe boundary, we obtain the following well-known fact:

Lemma 2.1. Let M be a hyperbolic 3-manifold with finitely generatedfundamental group. There is an exhaustion of M by a nested sequenceof submanifolds Ki such that:

(1) The boundary ∂Ki is smooth and strictly convex. Here, strictlyconvex means that 〈∇Xν,X〉 > 0, where ∇ is the Levi-Civita


connection, X is a vector tangent to ∂Ki and ν is the outernormal field along ∂Ki.

(2) Every convex cocompact end of M has a neighborhood disjointof Ki.

(3) The inclusion of Ki into M is a homotopy equivalence.

Continuing with the same notation as in Lemma 2.1, convexity im-plies that there is a well-defined map κKi

: M → Ki that takes a pointin M to the point in Ki closest to it. Strict convexity implies that thepreimage of a point x ∈ ∂Ki under this projection is a geodesic ray. Itfollows that the map

M \Ki → ∂Ki × (0,∞), x 7→ (κKi(x), d(x, κKi

(x))

is a diffeomorphism; in fact, its inverse is the radial coordinate map

∂Ki × (0,∞) →M \Ki, (x, t) 7→ expx(tν(x)),

where ν is the outer unit normal vector-field along ∂Ki.An end E which is not convex cocompact is said to be degener-

ate. It follows from the Tameness Theorem and earlier work of Bona-hon [Bon86] and Canary [Can93] that degenerate ends have very well-behaved geometry. For instance, every degenerate end E has a neighbor-hood which is completely contained in the convex core CC(M). Fromour point of view, the most important fact about degenerate ends isthe Thurston-Canary [Can96] covering theorem, of which we state thefollowing weaker version:

Covering theorem (Thurston-Canary). Let M and N be non-compacthyperbolic 3-manifolds, assume that M has finitely generated fundamen-tal group and no cusps, let π : M → N be a Riemannian cover and letE be a degenerate end of M . Then E has a standard neighborhood UEsuch that the restriction

π|UE : UE → π(UE)

of the covering π to UE is a covering map onto a standard neighborhoodof a degenerate end E ′ of N . More precisely, there is a finite coveringσ : ∂E → ∂E ′ and homeomorphisms

φ : ∂E × R → UE , ψ : ∂E ′ × R → π(UE)

with (ψ−1 π|UE φ)(x, t) = (σ(x), t). In particular, the covering π|UEhas finite degree.

Combining the Tameness theorem, Lemma 2.1 and the Covering the-orem we obtain:

7

Proposition 2.2. Let M and N be hyperbolic 3-manifolds with infinitevolume, assume that M has finitely generated fundamental group andno cusps, and let π : M → N be a Riemannian cover. Then M admitsan exhaustion by nested standard compact cores Ci ⊂ Ci+1 such that thefollowing holds:

(1) If a component S of ∂Ci faces a convex cocompact end of Mthen S is smooth and strictly convex.

(2) If a component S of ∂Ci faces a degenerate end of M then therestriction π|S : S → π(S) is a finite covering onto an embeddedsurface in N .

2.4. Conformal boundaries and Ahlfors-Bers theory. Continuingwith the same notation as in the previous section, we have a hyperbolic3-manifold M = H3/Γ with finitely generated fundamental group andno cusps.

Recall that the group Γ acts on H3 by isometries, and that its limitset, written Λ(Γ), is the closure of the set of fixed points in S2

∞ = ∂H3

of hyperbolic elements of Γ. The complement of Λ(Γ) is the domainof discontinuity Ω(Γ) = S2

∞ \ Λ(Γ), which is the largest open subsetof S2

∞ on which Γ acts properly discontinuously. In fact, Γ acts prop-erly discontinuously on H3 ∪Ω(Γ), and the quotient is a manifold withboundary M having interior M .

The action of Γ on S2∞ is by Mobius transformations, so ∂M inherits

a natural conformal structure and is accordingly called the conformalboundary of M . Its structure is closely tied with the geometry of M :for instance, the unique hyperbolic metric on ∂M compatible with thisconformal structure, called the Poincare metric, is similar to the in-trinsic metric on ∂CC(M). Specifically, the closest point projectionκ : M → CC(M) extends continuously to a map κ : ∂M → ∂CC(M),and we have the following theorem of Canary:

Proposition 2.3 (Canary [Can91]). For every ε > 0 there exists K >0 so that the following holds. Let M be a hyperbolic 3-manifold withfinitely generated fundamental group, such that every component of ∂Mhas injectivity radius at least ε in the Poincare metric. Then the closestpoint projection κ : ∂M → ∂CC(M) is K-lipschitz, where ∂M hasthe Poincare metric and ∂CC(M) is considered with the Riemannianmetric induced by its inclusion into M .

The conformal boundary plays an important role in the deformationtheory of hyperbolic 3-manifolds; in particular, a convex-cocompact hy-perbolic 3-manifold is determined up to isometry by its topology andconformal boundary. A more precise statement of this can be derived as


follows. Let M be the interior of a compact hyperbolizable 3-manifoldM in which each boundary component has negative Euler characteristic.Define CH(M) to be the set of all convex-cocompact hyperbolic metricson M , where two metrics are identified if they differ by an isometry iso-topic to the identity map. It follows from Thurston’s hyperbolizationtheorem [Thu86A] that CH(M) is nonempty, and it inherits a natu-ral complex structure through its relation to the representation varietyHom(π1(M),PSL2 C) (see [MT98, Section 4.3]). Then we have:

Theorem 2.4 (Ahlfors-Bers Parameterization, see [MT98]). The mapCH(M) → T (∂M), induced from the map taking a convex-cocompactuniformization of M to its conformal boundary, is a biholomorphicequivalence. Therefore, CH(M) is a complex manifold of dimension

−3

2χ(∂M) = −3χ(M).

The space CH(M) has a convenient naturality with respect to certaincoverings. Specifically, if M ′ is a cover of M with finitely generatedfundamental group then tameness and Canary’s covering theorem implythat convex-cocompact metrics on M lift to convex-cocompact metricson M ′. In fact,

Lemma 2.5. If M ′ is a 3-manifold with finitely generated fundamentalgroup and τ : M ′ →M is a covering map, there is a holomorphic map

τ ∗ : CH(M) → CH(M ′)

induced by the map taking a hyperbolic structure on M to its pullbackunder τ .

The point of the holomorphy is that τ ∗ is related to the holomor-phic map Hom(π1(M

′),PSL2 C) → Hom(π1(M),PSL2 C) defined byprecomposition with τ∗ : π1(M

′) → π1(M).

2.5. Geometric convergence. Recall that a sequence (Gi) of closedsubgroups of PSL2 C converges geometrically to a subgroup G if it doesin the Chabauty topology. More concretely, (Gi) converges geometri-cally to G if G is the subgroup of PSL2 C consisting precisely of thoseelements g ∈ PSL2 C such that there are gi ∈ Gi with gi → g in PSL2 C.In other words, G is the accumulation set of the groups Gi.

Most of our arguments are based on an interpretation of geometricconvergence in terms of the quotient manifolds H3/Gi.

Definition. A sequence (Mi, ωi) of pointed hyperbolic 3-manifolds con-verges geometrically to a pointed manifold (M∞, ω∞) if for every K ⊂M∞ compact with ω∞ ∈ K there is a sequence φi : K → Mi of smooth

9

maps with φi(ω∞) = ωi converging in the Ck-topology to an isometricembedding for all k ∈ N. We will refer to the maps φi as the almostisometric embeddings provided by geometric convergence.

Remark. Note that although the phrase ’converging in the Ck-topologyto an isometric embedding’ is suggestive and pleasing to the ear, it hasno meaning. One way to formalize this would be to say that for eachpoint x ∈ K there are isometric embeddings

βi : B(φi(x), ε) → H3

from ε-balls around φi(x) ∈Mi so that βi φi converges to an isometricembedding of some neighborhood of x ∈M∞ into H3.

Recall that by our convention above, a pointed hyperbolic manifoldis a manifold together with a base frame and that choosing a base frameωH3 of hyperbolic space we obtain a bijection between the sets of discretetorsion free subgroups of PSL2 C and of pointed hyperbolic 3-manifolds.Under this identification, the notions of geometric convergence of groupsand manifolds are equivalent (see for instance [BP92, Kap01]):

Proposition 2.6. Let G1, G2, . . . , G∞ be discrete and torsion-free sub-groups of PSL2 C and consider for all i = 1, . . . ,∞ the pointed hyper-bolic 3-manifold (Mi, ωi) where Mi = H3/Gi and ωi is the projectionof the frame ωH3 of H3. The groups Gi converge geometrically to G∞if and only if the pointed manifolds (Mi, ωi) converge geometrically to(M∞, ω∞).

2.6. Algebraic convergence. Let Γ be a finitely generated group.Recall that a sequence (ρi) of representations ρi : Γ → PSL2 C convergesalgebraically to a representation ρ if for every γ ∈ Γ we have ρi(γ) →ρ(γ) in PSL2 C. Jorgensen proved in [Jor76] that if each image ρi(Γ) isdiscrete and non-elementary then the same is true of ρ(Γ). His argumentalso shows that torsion cannot suddenly appear in the limit, so we havethe following theorem:

Proposition 2.7. Let Γ be a finitely generated group. Then the sub-set D(Γ) ⊂ Hom(Γ,PSL2 C) consisting of representations with discrete,torsion free and non-elementary image is closed with respect to the al-gebraic topology.

When a sequence (ρi) converges both algebraically to a representationρ and geometrically to some group G, it is easy to see that ρ(Γ) ⊂G. In other words, the manifold H3/ρ(Γ) covers the manifold H3/G.In particular, given a compact subset C ⊂ H3/ρ(Γ) we can projectit down to H3/G and then map the image to Mi under the almost


isometric embeddings given by geometric convergence. This producesmaps C →Mi = H3/ρi(Γ) which look more and more like the restrictionof a covering to C.

More generally, assume that H is a finitely generated subgroup of Gcontaining ρ(Γ). By the tameness theorem, the manifold H3/H containsa standard compact core CH . Composing the restriction to CH of thecovering H3/H → H3/G with the almost isometric embeddings given bygeometric convergence, we obtain for sufficiently large i maps CH →Mi

similar to those described above. Using the induced homomorphismsH → π1(Mi, ωi) one can then construct a sequence of representationsσi : H → PSL2 C converging algebraically to the inclusion ofH → G →PSL2 C. The assumption that ρ(Γ) ⊂ H implies then that σi(H) =ρi(Γ) for all i. In particular we deduce (compare with [Mc99, Lemma4.4]):

Proposition 2.8. Let Γ be a finitely generated group, (ρi) a sequence inD(Γ) converging algebraically to a representation ρ∞ and geometricallyto a group G ⊂ PSL2 C. If H ⊂ G is a finitely generated subgroup ofG containing ρ(Γ) then there is a sequence of representations σi : H →PSL2 C converging to the inclusion H → PSL2 C with σi(H) = ρi(Γ)for all sufficiently large i. In particular, the groups σi(H) convergegeometrically to G.

2.7. Roots. Recall that the elements in PSL2 C are either hyperbolic,parabolic or elliptic depending on their dynamical behaviour. Everyhyperbolic element γ ∈ PSL2 C stabilizes a geodesic A in H3, and ifα ∈ PSL2 C is a k-th root of γ, i.e. γ = αk, then αA = A. It followseasily that the set of k-th roots of γ is finite. A similar argument appliesin the parabolic and elliptic case, so we obtain the following well-known,and in this paper surprisingly important, fact:

Lemma 2.9. For every k ∈ Z and nontrivial element γ ∈ PSL2 C, theset α ∈ PSL2 C | αk = γ is finite.

Lemma 2.9 has the following immediate consequence:

Corollary 2.10. Let Γ ⊂ Γ′ be groups and assume that Γ′ contains asubgroup H such that (1) Γ and H generate Γ′ and (2) Γ∩H has finiteindex in H. Then for every faithful representation ρ : Γ → PSL2 C, theset ρ′ : Γ′ → PSL2 C | ρ′|Γ = ρ|Γ is finite.

Also, Theorem 1.3 from the introduction follows almost directly fromLemma 2.9.

Theorem 1.3 (Anderson). Assume that Γ is a finitely generated groupand (ρi) is a sequence of faithful representations in D(Γ) that converges

11

algebraically to a representation ρ and geometrically to a group G. Thenmaximal cyclic subgroups of ρ(Γ) are maximally cyclic in G. In par-ticular, if the image ρ(Γ) of the algebraic limit has finite index in thegeometric limit G, then ρ(Γ) = G.

Proof. If some maximal cyclic subgroup of ρ(Γ) is not maximally cyclicin G, then there is some g ∈ G \ ρ(Γ) that powers into it. So, gk =ρ(η) for some η ∈ Γ and k ≥ 2. Since g ∈ G there are γi ∈ Γ withlimi→∞ ρi(γi) = g. Taking powers we have then

limi→∞

ρi(γki ) = gk = ρ(η) = lim

i→∞ρi(η).

It follows from the Margulis lemma that ρi(γki ) = ρi(η) for sufficiently

large i. Since the representations ρi are faithful, this implies that γki = η

for large i. But the group Γ embeds in PSL2 C, by say ρ7, so Lemma2.9 shows that each of its elements has only finitely many kth-roots.This implies that up to passing to a subsequence we may assume thatγi = γj for all pairs (i, j), and hence g belongs to the algebraic limit.This is a contradiction, so the claim follows.

We included the proof of Theorem 1.3 here because its failure fornon-faithful sequences is the core of the examples presented in Section3.

2.8. Maximal Cusps. Finally, we describe a class of hyperbolic man-ifolds that we will use as building blocks in constructing our examplesin Section 3. A good reference for this section is [CCHS].

Assume that M is a compact, orientable, irreducible and atoroidal3-manifold with interior N and no torus boundary components. If Nhas a geometrically finite hyperbolic metric, then there is a collectionC of disjoint simple closed curves in ∂M such that

N ∼= M \ ∪γ∈Cγ,

by a homeomorphism whose restriction to N is isotopic to the identitymap. The curves in C are then determined up to isotopy, and corre-spond to the rank one cusps in N . We will say that the collection C hasbeen pinched. Each curve in C is homotopically nontrivial in M andno two curves are freely homotopic in M . It follows from Thurston’sHyperbolization Theorem [Kap01, Ota98] that any collection of curveson ∂M satisfying these two topological properties can be obtained asabove from a hyperbolic structure on N ; it is therefore said that sucha collection is pinchable.

One says that a component S ⊂ ∂M is maximally cusped if C con-tains a pants decomposition for S. In this case, any component of∂CC(N) that faces S is a totally geodesic hyperbolic thrice-punctured


sphere. If we have two hyperbolic 3-manifolds with maximally cuspedends, we can topologically glue their convex cores together along thesethrice-punctured spheres. Moreover, every homeomorphism betweenhyperbolic thrice punctured spheres can be isotoped to an isometry, soby altering the identifications we can ensure that our gluing produces ahyperbolic 3-manifold. Here is a precise description of the result of thisprocess.

Lemma 2.11 (Gluing Maximal Cusps). Let M1,M2 be compact, ori-entable 3-manifolds with interiors Ni that possess geometrically finitehyperbolic metrics. Assume that Si are maximally cusped componentsof ∂Mi with pinched pants decompositions Pi ⊂ Si. Then if h : S1 → S2

is a homeomorphism with h(P1) = P2, there is a hyperbolic 3-manifoldN with the following properties:

• N ∼= (M1\P1)th(M2\P2), so N has a rank 2 cusp correspondingto each element of P1 (or P2)

• N is the union of two disjoint subspaces isometric to N1\E1 andN2 \ E2, where Ei are the components of Ni \ CC(Ni) adjacentto Si.

3. Examples

In this section we construct the examples mentioned in the introduc-tion. All our constructions follow the same general strategy. Fixingsome group Γ, we start with a sequence of discrete representations (σi)

in D(Γ) that converges algebraically to a faithful representation σ∞whose image σ∞(Γ) has no parabolics and is the geometric limit of

σi(Γ). We find then a proper subgroup Γ ⊂ Γ with σi(Γ) = σi(Γ) forall i ∈ N and consider ρi = σi|Γ, the restriction of σi to the subgroup Γ.By construction, ρi converges algebraically to ρ∞ = σ∞|Γ and geomet-

rically to σ∞(Γ). Since Γ is a proper subgroup of Γ and σ∞ is faithful,we obtain that the algebraic limit ρ∞(Γ) = σ∞(Γ) is a proper subgroup

of the geometric limit σ∞(Γ).

Example 1. Let Γ be the fundamental group of a closed orientable sur-face of genus 3. There is a sequence of representations (ρi) in D(Γ) thatconverges algebraically to a faithful representation ρ and geometricallyto a group G such that

• G has no parabolics• ρ(Γ) has index 2 in G.

It will be apparent from the proof that there is nothing very specialabout genus 3. In fact, the argument will work whenever Γ is the

13

fundamental group of a closed orientable surface that nontrivially coversanother such surface.

A Sequence of Convex-Cocompact Handlebodies: Let (H,S) bea genus g handlebody with boundary surface S. Recall that the MasurdomainOH ⊂ PML(S) is the set of all projective measured laminationson S that intersect positively with every element of PML(S) that isa limit of meridians. Since it is open and the attracting laminationsof pseudo-anosov elements of Mod(S) are dense in PML(S), we maychoose a pseudo-anosov diffeomorphism f : S → S with attractinglamination λ ∈ OH . Theorem 2.4 implies that the deformation spaceof convex-cocompact hyperbolic metrics on a hyperbolizable 3-manifoldis parameterized by the Teichmuller space of its boundary. So, we canproduce a sequence of convex-cocompact metrics on H correspondingto an orbit of f on T (S). Concretely, after fixing some base conformalstructure on S, we have convex-cocompact hyperbolic manifolds Ni, i =1, 2, . . . and homeomorphisms hi : (H,S) → (Ni, ∂Ni) such that hi f i :S → ∂Ni is conformal.

Good Markings: By Proposition 2.3, the nearest point projectionηi : ∂Ni → ∂CC(Ni) is K-lipschitz for some constant K independentof i. Considering S with its Poincare metric, the map

σi := ηi hi f−i : S → Ni

is K-lipschitz as well. Since it is π1-surjective, we may use σi to markthe fundamental group ofNi with π1(S). Specifically, after taking a basepoint p ∈ S, the surjections (σi)∗ : π1(S, p) → π1(Ni, σi(p)) determineup to conjugacy a sequence of representations

ρi : π1(S, p) → PSL(2,C), H3/ρi(π1(S, p)) = Ni.

Since all σi are uniformly lipschitz, we may assume after passing toa subsequence that

ρi → ρ∞ : π1(S, p) → PSL(2,C)

in the algebraic topology. Set N∞ = H3/ρ∞(π1(S, p)).

Characterization of the Limit: The goal here is to verify thefollowing description of the algebraic limit ρ∞.

The representation ρ∞ : π1(S, p) → PSL(2,C) is a stronglimit of the sequence (ρi). It is faithful and purely lox-odromic, so N∞ ∼= Σg × R and has no cusps. One ofthe ends of N∞ is convex-cocompact and the other isdegenerate.


N1 genus g handlebody

N2 genus g handlebody

N 8 ≅ ΣgX

CC(N1)

CC(N

2)

σ1(p) σ2(p) σ 8(p). . .

... Σ2 X [0,1]

CC(N1)≅ σ1(S)6

handlebody

lengthincreaseswith i

Figure 1. A schematic description of the convergenceNi → N∞.

We first need to show that the marking of π1(N∞) given by ρ isinduced by an embedding σ∞ : S → N∞. Assume by passing to asubsequence that ρi(π1(S, p)) → G ⊂ PSL(2,C) geometrically, and letNG = H3/G. If base points are chosen for Ni on ∂CC(Ni), the resultingbased manifolds converge in the Gromov Hausdorff topology to NG.Our marking maps σi converge to a map σG : S → NG (the existenceof the limit can be shown by applying Arzela-Ascoli’s Theorem to thecomposition of σi with a sequence of almost isometric maps from Ni toNG coming from geometric convergence). There is a natural coveringmap π : N∞ → NG corresponding to the inclusion ρ∞(π1(S, p)) ⊂ G,and σG lifts to a map σρ : S → N∞ inducing the marking given by ρ∞.

Claim. σ∞ is an embedding, and its image bounds a neighborhood E ′

of a convex-cocompact end of N∞. Furthermore, E ′ ∼= Σg × R andπ|E′ : E ′ → NG is an embedding.

Proof. We will show that the image of σG bounds a convex-cocompactend in NG homeomorphic to Σg × R. It will follow immediately thatthis end lifts to an end E ′ as desired.

Let K ⊂ NG be a Gromov-Hausdorff limit of the sequence of convexcores CC(Ni), passing to another subsequence as necessary. Note thatK is convex and contains CC(NG). Its boundary ∂K is the limit of theboundaries ∂CC(Ni), and is therefore the image of σG. Using convexity,it is then not hard to see that σG is an embedding. Setting E = NG\K,we obtain a convex-cocompact end of NG. As ∂E ∼= Σg, the nearestpoint retraction gives a homeomorphism NG \K ∼= Σg × (0,∞).

15

Recall that the sequence (Ni) is marked by precomposing a home-omorphism hi : S → ∂Ni that extends to the handlebodies with apseudo-anosov map f : S → S that has attracting lamination λ ∈ OH .The following claim is the reason we chose λ in the Masur domain OH .

Claim. ρ∞ is faithful.

Proof. An equivalent statement of the claim is that the embedding σ∞ :S → N∞ is π1-injective. So by the Loop Theorem, it suffices to checkthat ρ∞ is injective on elements of π1(S, p) representable by simple loopson S.

Let γ ⊂ S be a simple closed curve. Then f i(γ) → λ in PML;since λ ∈ OH , which is an open subset of PML, for large i we havef i(γ) ∈ OH as well. In particular, f i(γ) is not compressible in H. Itfollows that

σi(γ) = ηi hi f i(γ)

is incompressible in Ni. Therefore ρi(γ) 6= Id for sufficiently large i. Astandard application of the Margulis Lemma shows that ρ∞(γ) 6= Id aswell (see, e.g.[MT98, Theorem 7.1]).

Geometrically, the reason that ρ∞ is faithful is that for large i allcompressible curves on ∂CC(Ni) are very long. Since a fixed generatingset for π1(S, p) maps under our markings to a set of loops on ∂CC(Ni)with uniformly bounded length, this implies that elements of π1(S, p)representing compressible curves in Ni must for large i be expressibleonly with very large words in the generators. So in the limit, there areno compressible curves.

As ρ∞ : π1(S, p) → PSL(2,C) is faithful, the Tameness Theoremimplies that N∞ ∼= Σg × R. From above, we know that one of the twotopological ends of N∞ is convex-cocompact. To analyze the geometryof the other end, we must first prove the following:

Claim. ρ∞(π1(S, p)) contains no parabolics elements.

The proof we give here follows an argument of McMullen. An al-ternative proof can be furnished by showing that the support of λ isunrealizable in N∞.

Proof. Recall that we have markings hi : H → Ni and that the followingdiagram commutes up to homotopy:

Sf

//

σi

S

σi+1

Ni

hi+1h−1i// Ni+1


By construction, hi f i : S → ∂N i is conformal. Therefore,

hi+1 h−1

i |S = (hi+1 f i+1) f (hi f i)−1|Sis quasiconformal with the same dilatation as f : S → S. Liftingto the universal cover, we obtain a sequence of quasiconformal mapsψi : S2

∞ → S2∞ with bounded dilatation that conjugate the action of

ρi f∗ : π1(S, p) → PSL(2,C) to that of ρi+1. (Here, f∗ can be any au-tomorphism of π1(S, p) that projects to the outer automorphism spec-ified by f ; a different choice will affect ψi by a composition with aMobius transformation.) Passing to a subsequence gives a quasicon-formal pointwise limit ψ that conjugates the action of ρ∞ f∗ to thatof ρ∞. So, if ρ∞(γ) is parabolic then ρ∞ f i

∗(γ) is parabolic for all i.Since f is pseudo-anosov no two of these iterates are conjugate. ButN∞ ∼= Σg × R can have only finitely many cusps, all of which must berank one, and therefore ρ∞ has only finitely many conjugacy classes ofprimitive parabolic elements. This is a contradiction unless ρ∞ has noparabolics.

It follows that each topological end of N∞ is either convex-cocompactor degenerate. As a corollary of Marden’s stability theorem [Mar74],which is actually a consequence of Theorem 4.3 below, we obtain:

Claim. N∞ has a degenerate end.

The next observation concludes our analysis.

Claim. ρi converges to ρ∞ strongly.

Proof. We need to show that the covering π : N∞ → NG is trivial.There is a neighborhood of the degenerate end of N∞ that is completelycontained in the convex core CC(N∞). So, N∞ \ E ′ is the union ofCC(N∞) and some compact set. A result of Thurston [KT90] andBonahon [Bon86] implies that the injectivity radius of N∞ is boundedabove inside of its convex core, so by extension there is an upper boundK for the injectivity radius of N∞ outside of E ′. Pick a point x ∈ NG

deep enough inside its convex-cocompact end so that inj(x,NG) > K.Then no element of N∞ \ E ′ can project to x. Since π is injective onE ′, |π−1(x)| = 1. Therefore π is trivial.

Restricting the Markings: Finally, we show that we can restrict ourrepresentations (ρi) to a subgroup of π1(S, p) to create a sequence withthe properties desired for our example.

Claim. After passing to a subsequence, there exists an index 2 subgroupΓ ⊂ π1(S, p) such that for each i > 0,

(σi)∗|Γ : Γ → π1(Ni(σi(p)))

17

is a surjection.

Proof. The proof consists of two simple observations. First, if M isa handlebody then there is an index 2 subgroup Γ ⊂ π1(∂M) thatsurjects onto π1(M). One can take, for example, the kernel of themap taking an element of π1(∂M) to its mod-2 intersection numberwith any longitude in a standard meridian-longitude basis for H1(∂M).Therefore, we can construct for each i and index 2 subgroup Γi with(σi)∗ : Γi → π1(Ni, σi(p)) a surjection. Since there are only finitelymany index 2 subgroups of π1(S, p), we can pass to a subsequence sothat a single Γ ⊂ π1(S, p) works for all i.

Consider now the sequence of representations ρi|Γ : Γ → PSL(2,C).The claim implies that ρi(Γ) = ρi(π1(S, p)), so

ρi(Γ) → ρ∞(π1(S, p))

geometrically. Now ρi → ρ∞ algebraically, so ρi|Γ → ρ∞|Γ. Therefore,since ρ∞ is faithful, ρ∞(Γ) has index 2 in ρ∞(π1(S, p)). We have there-fore provided a sequence of representations converging algebraically toa faithful representation whose image is an index 2 subgroup of the geo-metric limit, and so that the geometric limit has no parabolics. Thisconcludes our example.

An Alternate Method: The difficult part of the example above isconstructing a sequence of hyperbolic 3-manifolds (Ni), each homeo-morphic to the interior of a genus g handlebody, that converges geo-metrically to a hyperbolic 3-manifold N∞ homeomorphic to Σg × R.Such a sequence can be assembled differently by working backwardsfrom the limit.

Fix a Bers slice X ⊂ AH(π1(Σg)). There is a marked hyperbolic 3-manifold N∞ ∈ ∂X with no cusps, one degenerate end and one convex-cocompact end. One can construct N∞, for instance, as the algebraiclimit of a sequence obtained by iterating a pseudo-anosov mapping classon X, [Bro01]. McMullen has shown, [Mc91], that any point on theboundary of a Bers slice can be approximated by one-sided maximalcusps. This means that there is a sequence of marked hyperbolic 3-manifolds Mi ∈ ∂X converging algebraically to N∞, where each Mi hasa maximally cusped end. As N∞ has no cusps, Theorem 1.1 implies thatthere are base points pi ∈Mi so that (Mi, pi) → (N∞, p∞) geometrically.Furthermore, pi can be chosen to lie on the component of ∂CC(Mi)facing the convex-cocompact end of Mi. Note that since the thrice-punctured sphere components of ∂CC(Mi) disappear in the geometriclimit, their distances to pi grow without bound.


CC(M

i)CC(H)

glue

fill cusps

rank 2 cusps

Ni

Ni genus g handlebody

Σ2 X (0,1) Σ2 X (0,1)Σ2 X (0,1)

handlebody

handlebody

Figure 2. Producing long handlebodies by gluing max-imal cusps.

Passing to a subsequence, we may assume that the pinched pantsdecompositions on the maximally cusped ends of Mi all have the sametopological type. In other words, we may choose a pants decompositionP ⊂ Σg and homeomorphisms

M i∼= (Σg × [0, 1]) \ (P × 0)

for all i. Pick an identification of Σg with the boundary of a genus ghandlebody H so that P is a pinchable collection of curves on ∂H; thelatter condition can be ensured, for instance, by composing any fixedidentification with a high power of a pseudo-anosov homeomorphismof ∂H whose attracting lamination lies in the Masur domain. Thisallows us to endow H with a geometrically finite hyperbolic metricin which P has been pinched. Then both ∂CC(H) and the bottomboundary components of CC(Mi) are identified with Σg \ P , so wemay use Lemma 2.11 to glue them together. This produces a sequenceof hyperbolic 3-manifolds N ′

i equipped with isometric embeddings ofCC(H) and the subset Ki ⊂ Mi that is the union of the convex coreof Mi and its convex-cocompact end; we will denote the inclusion ofthe latter by ιi : Ki → N ′

i . Since the frontier of Ki in Mi consists ofthe thrice-punctured sphere components of ∂CC(Mi), its distance tothe base point pi ∈ Mi tends to infinity with i. The same holds forthe distances from ιi(pi) to the frontier of ιi(Ki) ⊂ N ′

i . Therefore, thesequence of based manifolds (N ′

i , ιi(pi)) converges geometrically to thesame limit, (N∞, p∞), as our original sequence.

19

Observe that N ′i is homeomorphic to the manifold obtained from

H by pushing P ⊂ ∂H into the interior of H and then drilling itout. The curves in P correspond to rank 2 cusps of N ′

i , so we maychoose (x, y)-coordinates for the Dehn filling space of each cusp so that(1, 0) corresponds to filling a curve that is contractible in H and (0, 1)represents filling a curve homotopic into P . Then the manifold N ′

i,n

obtained from (1, n)-Dehn filling on each cusp of N ′i is homeomorphic

to H (compare with [KT90, Section 3]). If n is large, an extension ofThurston’s Dehn filling theorem due to to Bonahon-Otal [BO88] andComar [Com96] implies that N ′

i,n is hyperbolic. Furthermore, there arebase points pi,n ∈ N ′

i,n such that (N ′i,n, pi,n) → (N ′

i , ιi(pi)) geometrically.An appropriate sequence (ni) can then be chosen so that (N ′

i,ni, pi,ni

)converges geometrically to (N∞, p∞). Setting Ni = N ′

i,nifinishes our

work.

Example 2. Let Γ be the fundamental group of a compression bodywith exterior boundary of genus 4 and connected interior boundary ofgenus 3. There is a sequence (ρi) in D(Γ) converging algebraically to arepresentation ρ and geometrically to a group G such that:

• G does not contain any parabolic elements.• ρ(Γ) has infinite index in G.

The representations here are constructed from those in the previousexample by using Klein-Maskit combination:

The Combination Theorem (see [MT98]). Let G1 and G2 be twodiscrete and torsion free subgroups of PSL2 C. Suppose that there existfundamental domains Di ⊂ Ω(Gi) for Gi, each containing the exteriorof the other. Then G = 〈G1, G2〉 is discrete, torsion free and is isomor-phic to G1 ? G2. Moreover, if the groups Gi do not contain parabolicsthen the same is true for G.

Recall our notation from the previous example: (ρi) is a sequence ofrepresentations in D(π1(Σ2)) converging strongly to a faithful represen-tation ρ∞ without parabolics, and Γ ⊂ π1(Σ2) is an index 2 subgroupwith ρi(Γ) = ρi(π1(Σ2) for all i. For convenience, set Gi = ρi(π1(Σ2))and G∞ = ρ∞(π1(Σ2)).

Our previous analysis implies that the convex cores of H3/Gi convergeto the convex core of H3/G∞ in the Gromov-Hausdorff topology, so thelimit sets Λ(Gi) converge to Λ(G∞) in the Hausdorff topology. SinceΩ(G∞) 6= ∅, it follows that there exist fundamental domainsDi ⊂ Ω(Gi)for the action of Gi converging to a fundamental domain D∞ for theaction of G∞ on Ω(G∞). Pick some loxodromic element α ∈ PSL(2,C)with fixed points contained in the interior of D∞. Moreover, assume


that its translation distance is large enough so that there is a fundamen-tal set for Ω(〈α〉) whose complement is entirely contained within D∞.After discarding a finite number of terms, its complement will also becontained in Di for all i. We now construct new representations

ρ′i : π1(Σ2) ? Z → PSL(2,C)

from ρi by sending 1 ∈ Z to α. If G′i ⊂ PSL(2,C) is the image of ρ′i,

the Klein-Maskit combination theorem implies that G′i = Gi ? 〈α〉 and

is discrete, torsion free and has no parabolics. The same statementsapply when i = ∞, showing in particular that ρ′i is faithful.

It follows, for instance, from the argument in [AC96B, Proposition10.2] that ρ′i → ρ′∞ strongly. Consider the subgroup Γ′ = Γ ? Z ⊂π1(Σ2) ? Z. Then ρ′i|Γ′ converges algebraically to ρ′∞|Γ′ . However, sinceρi(Γ) = ρi(π1(Σ2)) for all i, ρ′i(Γ

′) = ρ′i(π1(Σ2) ? Z) as well. So, ρ′i(Γ′)

converges geometrically to G′∞. Since ρ′∞ is faithful, ρ′∞(Γ′) has infinite

index in G′∞. Therefore ρ′i : Γ′ → PSL(2,C) is a sequence of represen-

tations for which the algebraic limit has infinite index in the geometriclimit. As mentioned above, the geometric limit Γ′ has no parabolics.Since Γ′ is isomorphic to the fundamental group of a compression bodywith exterior boundary of genus 4 and connected interior boundary ofgenus 3, we have provided the desired example.

The geometry of Example 2: Here is a schematic picture of theabove example, on the level of quotient manifolds.

N2 N1 N 8

CC(N1)

CC(N

2) ... compression body

handlebody

compression body

compressionbody

Σ2 X (0, 8)

genus 3handlebody

genus 3handlebody ’’’

Figure 3. The geometric convergence N ′i → N ′

∞.

The manifolds N ′i = H3/G′

i are all convex-cocompact hyperbolic 3-manifolds homeomorphic to the interior of a genus 3 handlebody. Thestrong limit N ′

∞ = H3/G′∞ is then a compression body with genus 3

exterior boundary and connected, genus 2 interior boundary. Its exte-rior end is convex-cocompact and its interior end is degenerate. After

21

restricting our representations to Γ′, the manifolds in our sequence andtheir geometric limit remain unchanged, but the algebraic limit is now acover of N ′

∞ homeomorphic to a compression body with genus 4 exteriorboundary and connected, genus 3 interior boundary.

Example 3. Let Γ be the fundamental group of a compression bodywith exterior boundary of genus 4 and connected interior boundary ofgenus 3. There is a sequence (ρi) in D(Γ) converging algebraically to arepresentation ρ and geometrically to a group G such that:

• ρ(Γ) has does not contain any parabolic elements.• G is infinitely generated.

In [Thu86B], Thurston exhibited a sequence of representations of aclosed surface group into PSL(2,C) converging geometrically to a groupthat is not finitely generated. However, the algebraic limit of theserepresentations contains parabolic elements. The idea here is to attachpieces of the handlebodies in Example 2 to the manifolds in his sequenceso that the parabolics are hidden from the algebraic limit.

To facilitate such a combination, we must build a variant of Example2 in which each of the handlebodies in the sequence is maximally cusped.Let M be a compression body with genus 3 exterior boundary SE andconnected, genus 2 interior boundary SI . Assume that PE ⊂ SE is apants decomposition consisting of curves in the Masur domain of M .

Claim. There is a sequence of maximally cusped pointed hyperbolic 3-manifolds (Ni), each homeomorphic to the interior of a genus 3 handle-body, that converges geometrically to a hyperbolic 3-manifold N∞ home-omorphic to the interior of M in which PE has been pinched. Moreover,the subsets CC(Ni) ⊂ Ni converge to CC(N∞) geometrically.

Proof. The sequence (Ni) will be constructed in two steps. First, wewill produce a sequence of hyperbolic 3-manifolds Mi homeomorphicto the interior of M in which both ends are maximally cusped. Thesequence will converge geometrically to the manifold N∞ referenced inthe statement of the claim. We will then use the same gluing trickexploited in Example 1 to cap off the interior ends of each Mi withoutchanging the sequence’s geometric limit, thus producing the desiredsequence of handlebodies (Ni).

Fix pants decompositions PE and PI for the boundary components ofM , and assume that every curve in PE lies in the Masur domain of M .If we choose a pseudo-anosov diffeomorphism f : SI → SI , then for eachi we have a new pants decompositions f i(PI) for SI . It is not hard tocheck that for each i, PE ∪ f i(PI) is a pinchable collection of curves on


∂M (see Section 2.8). So, there is a sequence of marked hyperbolic 3-manifolds Mi ∈ AH(M) in which PE ∪ f i(PI) have been pinched. Notethat in fact Mi lies in the deformation space AH(M,PE) of hyperbolicstructures on the interior of M in which the curves of PE representparabolics. Since PE consists of curves lying in the Masur domain, thepared manifold (M,PE) has incompressible and acylindrical boundary.It follows from a theorem of Thurston, [Thu86A, Theorem 7.1], thatAH(M,PE) is compact. So after passing to a subsequence, we mayassume that (Mi) converges algebraically to some N∞ ∈ AH(M,PE).

We claim that the only parabolic loops in N∞ are those that are freelyhomotopic into PE. The end of N∞ facing SE is maximally cusped byPE, so there is no room for additional cusps there. It therefore suffices toshow that the end facing SI has no cusps. Work of Thurston, Bonahonand Brock, implies that there is a continuous map

length : AH(M)×ML(SI) → R

that extends the function that assigns to an element N ∈ AH(M) and asimple closed curve γ ∈ SI the shortest length of a curve inN homotopicto γ (see [Bro01]). Since lengthMi

(f i(PI)) = 0 for all i, in the limit wehave lengthM∞(λ) = 0, where λ is the attracting lamination of f . Thisimplies that λ cannot be geodesically realized by a pleated surface inN∞homotopic to SI . Let M∞ be the cover of N∞ corresponding to π1(SI).

The end of N∞ facing SI lifts homeomorphically to an end E of N∞.The other end of N∞ has no cusps and is therefore convex-cocompactby the Tameness Theorem and Thurston’s covering theorem. Since λ isfilling and unrealizable in N∞, the argument in [MT98, Theorem 6.34]then shows that E is degenerate with ending lamination λ. In particular,E has no cusps. Projecting down, the same is true for the end of N∞facing SI .

This shows that the convergence Mi → N∞ is type preserving, soTheorem 1.1 implies that the convergence is strong. So, base frames forMi can be chosen so that the sequence converges geometrically to N∞.The rest of the argument follows that given at the end of Example 1. Fixa genus 2 handlebodyH and choose a hyperbolic metric on its interior inwhich a pants decomposition P ⊂ ∂H with the same topological type asPI ⊂ SI has been pinched. We can then create for each i a hyperbolic3-manifold N ′

i by removing from Mi the component of Mi \ CC(Mi)facing SI and gluing CC(H) in its place. As in Example 1, the sequence(N ′

i) converges geometrically to N∞. Performing an appropriate Dehnfilling on each N ′

i yields a sequence of hyperbolic 3-manifolds (Ni),each homeomorphic to the interior of a genus 3 handlebody, that alsoconverges geometrically to N∞.

23

We must show that CC(Ni) → CC(N∞) geometrically. First, ev-ery component of ∂CC(N∞) is contained in a geometric limit of somesequence of components of ∂CC(Ni). These are all thrice puncturedspheres, however, so in fact we have that for large i there are componentsof ∂CC(Ni) that closely approximate each component of ∂CC(N∞).However, ∂CC(Ni) and ∂CC(N∞) both have 6 components, so forlarge i they must almost coincide. From this, it is easy to check thatCC(Ni) → CC(N∞) geometrically.

Fix a geometrically finite hyperbolic structure on Σ3 × R in whichboth ends are maximally cusped, and let C be its convex core. Thenthere are pants decompositions P+,− ⊂ Σ3 so that

C ∼= (Σ3 × [−1,+1]) \ (P− × −1 ∪ P+ × +1),

and we label the components of ∂C as positive and negative accordingly,so that ∂C = ∂+C ∪ ∂−C. Let E be the component of the complementof C that faces its positive boundary components. Assume that thepairs (Σ3, P+,−) and (SE, PE) have the same topological type, and thatevery curve in P+ intersects some curve in P−.

We now glue these pieces to the manifolds Ni and N∞ from theprevious Lemma. To begin with, let

N ′∞ = CC(N∞) th C tg C tg · · · .

Here, the gluing maps h : ∂CC(N∞) → ∂−C and g : ∂−C → ∂+Ccan be any isometries that extend to maps (SE, PE) → (Σ3, P−) and(Σ3, P+) → (Σ3, P−). This ensures that N ′

∞ is constructed as in Lemma2.11. Note that the inclusion map CC(N∞) → N ′

∞ is π1-injective.Next, for large i geometric convergence and the gluing map h deter-

mine an identification hi : ∂CC(Ni) → ∂−C. Define

N ′i = CC(Ni) thi

C tg · · · tg C︸︷︷︸i times

tidE.

Recall that CC(Ni) → CC(N∞) geometrically. From this it follows thatif N ′

i is given the base frame of Ni produced in the previous Lemma,then (N ′

i) converges geometrically N ′∞.

As in Example 1, performing (1, n)-Dehn filling on each of the cusps inN ′

i produces, for large n, a new hyperbolic manifold N ′i,n homeomorphic

to the interior of a genus 3 handlebody. Also, an appropriate diagonalsequence N∗

i = N ′i,ni

can be chosen to converge geometrically to N ′∞.

In summary, we have proven the following claim.


compression body

handlebody

...

Σ 3 X

(0,1

)

Σ 3 X

(0,1

)

Σ 2 X

(0,1

)

Σ 3 X

(0, 1

)

glue and fill cusps

length increases

with i

i copies

Figure 4. The construction of the manifold N∗i .

Claim. There is a sequence of convex-cocompact pointed hyperbolic 3-manifolds N∗

i , each homeomorphic to the interior of a genus 3 handle-body, that converges geometrically to a hyperbolic 3-manifold N ′

∞ withinfinitely generated fundamental group.

We now obtain the sequence of representations advertised in the state-ment of this example by marking the manifoldsN∗

i appropriately. Recallthat the fundamental group of the compression body M splits as a freeproduct

π1(M) = π1(SI) ? 〈α〉 ∼= π1(Σ2) ? Z,for some element α ∈ π1(M). The inclusion map M ∼= CC(N∞) → N ′

∞is π1-injective, so it determines an embedding

ρ∞ : π1(Σ2) ? Z → π1(N′∞).

Then ρ∞ identifies a finite generating set for π1(Σ2)?Z with a finite setof loops in N ′

∞. For large i, geometric convergence provides an almostisometric embedding of these loops into N∗

i ; therefore, there are inducedhomomorphisms

ρi : π1(Σ2) ? Z → π1(N∗i ).

In fact, ρi is surjective, and therefore is a marking of π1(N∗i ).

The sequence of marked hyperbolic manifolds (N∗i , ρi) converges al-

gebraically to the cover of N ′∞ corresponding to the image of ρ∞, and

converges geometrically to N ′∞ (as noted above). Note that π1(N

′∞) is

not finitely generated. We are not quite done, however, because thealgebraic limit here has cusps. To hide the cusps, we will use the fi-nite index trick exploited in Example 1, but for this to work the pantsdecomposition PE used above must be chosen more carefully.

Claim. There is a pants decomposition PE ⊂ SE consisting of curvesin the Masur domain, none of which are conjugate into a subgroup ofπ1(M) of the form Γ ? 〈α〉, where Γ < π1(SI , p) has index 2.

25

Deferring the proof for a moment, pick as in Examples 1 and 2 anindex 2 subgroup Γ ⊂ π1(Σ2) so that ρi|Γ?Z surjects onto π1(N

∗i ). If in

constructing (N∗i ), the pants decomposition PE is chosen as indicated

in the above claim, there will be no parabolics in the algebraic limit of(ρi|Γ?Z). However, since this is still a sequence of markings for N∗

i , thegeometric limit will be N ′

∞, which has infinitely generated fundamentalgroup. This finishes the example.

Proof of Claim. Although the claim is purely topological, the proof wegive uses 3-dimensional hyperbolic geometry. It would be nice to give amore straightforward proof; also, it is possible that the the second partof the conclusion is satisfied by any pants decomposition of curves inthe Masur domain.

To begin, construct by some means a hyperbolic manifold N homeo-morphic to the interior of M that has no cusps and in which both endsare degenerate. One way to produce N is as follows. First, find a hy-perbolic manifold homeomorphic to Σ3×R that has one degenerate endand one maximally cusped end. This can be done using an argumentsimilar to the construction of N∞ above. We can then glue its convexcore to the convex core of N∞ as in Lemma 2.11 and appropriately fillthe resulting cusps to create a totally degenerate hyperbolic manifoldN homeomorphic to the interior of M .

Fix an index 2 subgroup Γ ⊂ π1(SI , p), and let NΓ be the cover of Ncorresponding to Γ? 〈α〉. Then NΓ has a degenerate end homeomorphicto Σ3 × [0,∞) that double covers the genus 2 end of N . Adjoining aloop in NΓ representing α to a level surfaces of this end and thickeningproduces a compact core for NΓ homeomorphic to the interior of acompression body with genus 4 exterior boundary and connected, genus3 interior boundary. The tameness theorem of Agol [Ago] and Calegari-Gabai [CG06] implies that NΓ is itself homeomorphic to the interior ofsuch a compression body, and a theorem of Canary [Can93] implies that

its genus 4 end, E , is either degenerate or convex-cocompact. It cannotbe that both ends of NΓ are degenerate, for then Canary’s coveringtheorem [Can96] would imply that Γ ? 〈α〉 is finite index in π1(N).

Therefore, E is convex-cocompact.Since the genus 3 end, E , of N is degenerate, it has an ending lami-

nation λ ⊂ SE. Canary has shown that λ lies in the Masur domain ofM (see [Can93] for a proof of this and the uniqueness of λ). Choose apants decomposition PE ⊂ SE consisting of curves that lie close to λ inPML(SE). The Masur domain of M is an open subset of PML(SE),so we may assume that each curve in PE lies inside of it. Furthermore,their geodesic representatives in N lie very deep inside of E . As E is


convex-cocompact, the convex core of NΓ covers a subset of N thathas bounded intersection with E . So, we may assume that the geodesicrepresentatives of curves in PE do not intersect its image. If some curvein PE were conjugate into Γ ? 〈α〉, then its geodesic representative inN would lift to a closed geodesic in NΓ. Every closed geodesic in NΓ iscontained in its convex core, so this is impossible.

4. Proof of Theorem 1.4

Before beginning the bulk of the proof, we will present a technicallemma whose proof requires a bit of differential geometry. Afterwards,Theorem 1.4 will follow from purely synthetic arguments.

Recall from Proposition 2.2 that a hyperbolic 3-manifold M withfinitely generated fundamental group and no cusps contains a compactcore C ⊂M for which each component of ∂C facing a convex-cocompactend of M is smooth and strictly convex. For convenience, let Scc bethe union of those components of ∂C facing convex cocompact endsand Ecc the union of the adjacent components of M \ C. Then Ecc ishomeomorphic to Scc × (0,∞) via ’radial coordinates’:

R : Scc × (0,∞) → Ecc, R(x, t) = expx(tν(x)),

where ν is the outer unit-normal vectorfield along Scc.If f : C → N is a smooth immersion into some complete hyperbolic

3-manifold N , it has a natural radial extension f : C ∪ Ecc → N .Namely, there is a radial coordinate map along the image of Scc:

Rf : Scc × (0,∞) → N, Rf (x, t) = expx(tνf (x)),

where νf (x) is the unit vector in TNf(x) orthogonal to dfx(TSx) thatpoints away from f(C), and one can then define

f(p) =

Rf R−1(p) p ∈ Ecc

f(p) p ∈ C.

Observe that f is continuous, and differentiable everywhere but on Scc.In the situations where we will find radial extensions useful, the map

f will be very close in the C2 topology to a Riemannian immersion. Inparticular, the (strict) convexity of the surface Scc will persist in theimage. This implies a convenient regularity in the radial extension:

Lemma 4.1. If f : C → N is a smooth immersion with f(Scc) convex,there exists some L > 0 so that for all p ∈ Ecc and v ∈ TMp,

1

L≤ ||dfp(v)||

||v||≤ L.

27

The following global statement comes from applying Lemma 4.1 anda compactness argument on C.

Corollary 4.2 (Radial Extensions are Locally Bilipschitz). If f : C →N is a smooth immersion with f(Scc) convex, there exists some L > 0so that every p ∈ C∪Ecc has a neighborhood on which f is L-bilipschitz.

Proof of Lemma 4.1. Consider a component E ⊂ Ecc, and let S ⊂ Scc

be the adjacent component of ∂C. Here, f is the composition Rf (R)−1

of radial coordinate maps, so to prove the Lemma it suffices to find aconstant L so that for all (x, t) ∈ S × (0,∞) and v ∈ TSx × R,

(4.1)1

L≤||(dRf )(x,t)(v)||||dR(x,t)(v)||

≤ L.

Since the ratio is one when v is contained in the R factor, from now onwe will assume v ∈ TSx.

We first estimate ‖dR(x,t)(v)‖. Given x ∈ S and v ∈ TSx, let g(s)be a curve in S with g(0) = x and g′(0) = v, and consider the geodesicvariation

γs(t) := R(g(s), t) = expg(s)(tν(g(s)))

The corresponding Jacobi field Jx,v(t) = ddsR(g(s), t)|s=0 along the geo-

desic γ0(t) then satisfies Jx,v(t) = dR(x,t)(v).The Jacobi-field Jx,v(t) is determined by its initial conditions

Jx,v(0) = dR(x,0)(v), and

∇dtJx,v(t)|t=0 =

∇dt

d

dsexpg(s)

(tν(g(s))

)|t=0

=∇ds

d

dtexpg(s)

(tν(g(s))

)|t=0

=∇dsν(g(s)) = ∇dR(x,0)(v)ν

Therefore, we have

Jx,v(t) = cosh(t)E1(t) + sinh(t)E2(t),

where E1(t) and E2(t) are the parallel vector fields along γ0 with E1(0) =dR(x,0)(v) and E2(0) = ∇dR(x,0)(v)ν. That the right-hand side satisfies

the Jacobi equation follows quickly from the fact that E1(t) and E2(t)are both orthogonal to γ′0(t).

The triangle inequality, together with the fact that the vector fieldsE1 and E2 have constant length, shows that

‖Jx,v(t)‖ ≤ cosh(t)(‖dR(x,0)(v)‖+ ‖∇dR(x,0)(v)ν‖).


On the other hand we have

‖Jx,v(t)‖2 = cosh(t)2‖dR(x,0)(v)‖2 + sinh(t)2‖∇dR(x,0)(v)ν‖2

+ 2 cosh(t) sinh(t)〈dR(x,0)v,∇dR(x,0)(v)ν〉By convexity of Scc, the last term in the sum is positive. A little bit ofalgebra and the fact that Jx,v(t) = dR(x,t)(v) yields

(4.2)sinh(t)

2≤

‖dR(x,t)(v)‖‖dR(x,0)(v)‖+ ‖∇dR(x,0)(v)ν‖

≤ cosh(t)

Since f(Scc) is convex, a similar computation shows

(4.3)sinh(t)

2≤

‖d(Rf )(x,t)(v)‖‖d(Rf )(x,t)(v)‖+ ‖∇d(Rf )(x,t)vνf‖

≤ cosh(t)

By compactness the ratio between the denominators in (4.2) and (4.3)is uniformly bounded from above and below. In other words, there issome positive constance c with

(4.4)sinh(t)

2c cosh(t)≤‖d(Rf )(x,t)(v)‖‖dR(x,t)(v)‖

≤ 2c cosh(t)

sinh(t)

for all (x, t) ∈ S × (0,∞) and v ∈ TxS. If t is constrained away fromzero then the lower and upper bounds in (4.4) are bounded by positivenumbers from below and above respectively. When t = 0 both dR(x,t)

and d(Rf )(x,t) have maximal rank, so constant positive bounds arisefrom a compactness argument. This yields (4.1) and concludes theproof of the Lemma.

We are now ready to prove the main result of this section.

Theorem 1.4. Let Γ be a finitely generated group and (ρi) a sequencein D(Γ). Assume that (ρi) converges algebraically to a representationρ ∈ D(Γ) and geometrically to a subgroup G of PSL2 C. If

• ρ(Γ) does not contain parabolic elements,• maximal cyclic subgroups of ρ(Γ) are maximal cyclic in G,

then G = ρ(Γ).

Before going further, set MA = H3/ρ(Γ), MG = H3/G and Mi =H3/ρi(Γ) for i = 1, 2, . . . , choose a base frame ωH3 for hyperbolic spaceH3 and let ωi, ωA and ωG be the corresponding base frames of Mi,MA and MG respectively. By Proposition 2.6, the pointed manifolds(Mi, ωi) converge geometrically to (MG, ωG). We may assume withoutloss of generality that MG has infinite volume. Otherwise, it is eithercompact, in which case the sequence (Mi) is eventually stable, or it isnoncompact and (Mi) is obtained by performing Dehn filling on MG

29

with larger and larger coefficients [BP92, Theorem E.2.4]. In the lattercase, it is not hard to see that there must be parabolics in the algebraiclimit ρ(Γ).

There is a covering map π : MA → MG induced by the inclusionρ(Γ) ⊂ G, and our goal is to show that this is a homeomorphism.Recall that Proposition 2.2 provides an exhaustion of MA by compactcores C ⊂MA such that

(1) if a component S of ∂C faces a convex cocompact end of MA

then S is smooth and strictly convex,(2) if a component S of ∂C faces a degenerate end of MA then the

restriction π|S : S → π(S) is a finite covering onto an embeddedsurface in MG.

Fixing a compact core C0 ⊂ MA, we may also assume that all C arelarge enough to contain C0 and satisfy the following property:

(3) π(C0)∩π(S) = ∅ for any component S ⊂ ∂C facing a degenerateend of MA.

Then to prove that π is a homeomorphism, it clearly suffices to showthat π|C is injective for all such compact cores C ⊂MA.

Fix a compact core C ⊂ MA as described above. The geometricconvergence (Mi, ωi) → (MG, ωG) supplies for sufficiently large i analmost isometric embedding φi : π(C) → Mi, so for large i, we have amap

fi : C →Mi, fi = φi πthat behaves much like the restriction of a nearly Riemannian coveringmap. In fact, fi is π1-surjective. For if S ⊂ Γ is a finite generating setthen C contains loops based at ωA representing the elements of ρ(S); fi

then maps these loops to loops in Mi representing ρi(S), which generateπ1(Mi) ∼= ρi(Γ). We aim to show that fi is actually an embedding, asthe same will then be true for π|C .

We first consider the case where MA is convex-cocompact, as theproof is particularly simple. In this case, every component of ∂C isstrictly convex and faces a convex-cocompact end of MA, so fi radiallyextends (as in the beginning of the section) to a globally defined map

fi : MA →Mi.

We claim that this is a covering map for i >> 0. To see this, notethat when i is large fi is C2-close to a local isometry, so the strictconvexity of ∂C persists after applying fi. Therefore Corollary 4.2applies to show that fi is (uniformly) locally bilipschitz. It is well-known that any locally isometric map between complete Riemannianmanifolds is a covering map, and in fact the same proof applies to


MA

MG

Mi

C

almost isometric embedding φi

fi

radially extend

PiS

PiS

π

chop off and replace with Pi

S

S

π(C)fi (S)

fi (C)

Figure 5. Extending fi = φi π to a cover of Mi.

locally bilipschitz maps. So, fi : MA →Mi is a covering map. However,fi is π1-surjective, so its extension fi is a π1-surjective covering map,and therefore a homeomorphism. This shows that fi is injective, andin particular π|C is as well.

In the general case, the argument needs modification because onecannot radially extend fi into the degenerate ends of MA. To deal withthis, we will alter the problematic parts of MA so that an extension offi is obvious.

Claim. If S ⊂ ∂C faces a degenerate end of MA, then the restrictionof fi to S is an embedding with image a separating surface in Mi.

Proof. Our choice of C ensures that π|Sdis a finite covering onto its

image. The assumption that maximally cyclic subgroups of ρ(Γ) aremaximally cyclic in G then implies that π|Sd

is an embedding, for oth-erwise there would be a loop in π(Sd) that does not lift to MA, but thathas a power which does.

Next, property (3) above implies that every component S ⊂ fi(Sd) isdisjoint from fi(C0). However, the argument given to show that fi|C isπ1-surjective also applies to fi|C0, so every loop in Mi is homotopic intofi(C0) and therefore has trivial algebraic intersection with S. ThereforeS is separating.

For each such S, let P Si be the closure of the component of Mi \fi(S)

containing fi(C0). Then if Ecc is the union of the components of MA\Cthat are neighborhoods of convex cocompact ends, one can construct anew 3-manifold

31

M ′A =

(C ∪ Ecc

)tfi

( ⋃S

P Si

)by gluing each P S

i to C ∪ Ecc along S. The map fi extends naturallyto a continuous map fi : M ′

A →Mi; the extension into Ecc is radial andon P S

i we use the natural inclusion into Mi. It is easy to see that fi isa π1-surjective covering map, so the proof ends the same way it did inthe previous case.

We would like to observe that we did not really use that every maxi-mal cyclic group in ρ(Γ) is maximal cyclic in G. We namely proved thefollowing less aesthetically pleasant but more general theorem:

Theorem 4.3. Let Γ be a finitely generated group and (ρi) a sequencein D(Γ). Assume that (ρi) converges algebraically to a representation ρand geometrically to a subgroup G of PSL2 C. If

• ρ(Γ) does not contain parabolic elements, and• every degenerate end of H3/ρ(Γ) has a neighborhood which em-

beds under the covering H3/ρ(Γ) → H3/G,

then G = ρ(Γ).

Before concluding this section, observe that Theorem 1.4 togetherwith Theorem 1.3 imply the Anderson-Canary Theorem 1.1 mentionedin the introduction.

5. Attaching roots

In this section we prove:

Proposition 5.1. Let M and N be hyperbolic 3-manifolds with infinitevolume and let τ : M → N be a covering. Assume that N has no cusps,that π1(M) is finitely generated and that M has a degenerate end whichdoes not embed under the covering τ . Then there is a hyperbolic 3-manifold M ′ with finitely generated fundamental group, with χ(M ′) >χ(M) and coverings τ ′ : M →M ′ and τ ′′ : M ′ → N with τ = τ ′′ τ ′.

Continuing with the notation above, by Proposition 2.2 the manifoldM has a standard compact core C with the property that if a componentS of ∂C faces a degenerate end of M then the restriction of τ to S is acovering onto an embedded surface in N . The assumption that M hasa degenerate end which does not embed under the covering τ impliesthat there is actually a component S0 of ∂C such that

τ |S0 : S0 → τ(S0)

is a non-trivial covering. Observe that by the covering theorem theembedded surface τ(S0) ⊂ N faces a degenerate end of N .


Choosing a base point ∗ ∈ S0, we set

Γ = π1(M, ∗) and H = π1(τ(S0), τ(∗)).The desired manifold M ′ will be the cover of N corresponding to thesubgroup

Γ′ = 〈τ∗(Γ), H〉 ⊂ π1(N, τ(∗)).By construction, π1(M

′) ∼= Γ′ is finitely generated and there are cover-ing maps τ ′ : M →M ′ and τ ′′ : M ′ → N with τ = τ ′′ τ ′, so it remainsonly to prove that χ(M ′) > χ(M).

For our purposes, the most useful way to interpret the Euler charac-teristic will be through its relation to the dimension of the deformationspaces CH(M) and CH(M ′) of convex-cocompact hyperbolic structureson M and M ′ (see Section 2.4). Observe that since M and M ′ havefinitely generated fundamental group and no cusps, they are homeomor-phic by the tameness theorem to the interiors of compact hyperbolizable3-manifolds M and M ′ whose boundary components have negative Eu-ler characteristic. It follows from Section 2.4 that CH(M) and CH(M ′)are complex manifolds of C-dimensions −3χ(M) and −3χ(M), respec-tively, and that there is a holomorphic map

(τ ′)∗ : CH(M ′) → CH(M)

defined by lifting hyperbolic structures using τ ′ : M → M ′. We willprove:

Claim. (τ ′)∗ has discrete fibers and is not open.

Since any holomorphic map with discrete fibers is open unless thedimension of the domain is smaller than the dimension of the image,we deduce from the claim that

−3χ(M ′) = dimC CH(M ′) < dimC CH(M) = −3χ(M).

Therefore, χ(M ′) > χ(M) and hence Proposition 5.1 will follow oncewe prove the claim.

The first part of the claim is almost immediate. If τ ′∗ : Γ → Γ′ is theinclusion induced by the covering τ ′ : M →M ′ and H < Γ′ is as above,then by construction

• Γ′ is generated by τ ′∗(Γ) and H• τ ′∗(Γ) ∩H has finite index in H.

It follows from Corollary 2.10 that a representation Γ → PSL2 C hasonly finitely many extensions to Γ′. Therefore, hyperbolic structureson M ′ that map under (τ ′)∗ to the same element of CH(M) have onlyfinitely many options for holonomy representations, up to conjugacy.However, the elements of CH(M ′) with holonomy in any fixed conjugacy

33

class form a discrete subset of CH(M ′) (see page 154, [MT98]), so (τ ′)∗

must have discrete fibers.To show that it is not open, we use the Ahlfors-Bers parameterization

to produce from (τ ′)∗ a holomorphic map

β : T (∂M ′) ∼= CH(M ′)(τ ′)∗

// CH(M) ∼= T (∂M).

The Teichmuller spaces of ∂M ′ and ∂M split as products of the Teich-muller spaces of their connected components; let S ⊂ ∂M be the com-ponent adjacent to the degenerate end that the surface S0 faces. Thecovering theorem implies that τ ′ extends to a nontrivial cover τ : S → S ′

onto some connected component S ′ ⊂ ∂M ′. With respect to the de-compositions

T (∂M) = T (S)× T (∂M \ S), T (∂M ′) = T (S ′)× T (∂M ′ \ S ′)the map β can be written as

β(σ1, σ2) = (τ ∗σ1, β(σ1, σ2))

where τ ∗ : T (S ′) → T (S) is the map induced by the covering τ :S → S ′. This covering is non-trivial, so the image of τ ∗ has positivecodimension and hence the same holds for the image of β. Therefore βis not open, implying the same for (τ ′)∗.

Remark. A purely homological computation yields that under the sameassumptions as in Proposition 5.1 we have b1(M

′) < b1(M) where b1(·)is the first Betti number with R-coefficients. However, this homologicalargument does not seem to work in the relative case that we will discussin section 7. This is why we choose to work with Euler characteristicsand deformation spaces instead.

6. Proof of Theorem 1.2

Recall the statement of Theorem 1.2:

Theorem 1.2. Let Γ be a finitely generated group and (ρi) a sequencein D(Γ). Assume that (ρi) is algebraically convergent and convergesgeometrically to a subgroup G of PSL2 C. If G does not contain parabolicelements, then G is finitely generated.

If the hyperbolic 3-manifold H3/G is compact then G is obviouslyfinitely generated. Assume from now on that this is not the case; sinceG does not contain parabolic elements, this assumption impies thatH3/G has infinite volume.

Among all finitely generated subgroups of G which contain ρ(Γ),choose H such that the associated hyperbolic 3-manifold H3/H has


maximal Euler characteristic χ(H3/H). Since H ⊂ G we have a cover-ing H3/H → H3/G. Using the maximality assumption, we obtain fromProposition 5.1 that every degenerate end of H3/H embeds under thiscover.

On the other hand, the assumption that H contains ρ(Γ) implies, byProposition 2.8, that there is a sequence of representations σi : H →PSL2 C converging algebraically to the inclusion of H in G such thatthe groups σi(H) converge geometrically to G. Theorem 4.3 impliesnow that H = G. In particular, G is finitely generated. This concludesthe proof of Theorem 1.2.

7. Parabolics

It is a well established fact that most theorems in the deformationtheory of Kleinian groups that hold in the absence of parabolics holdalso, in some probably weaker form, in the presence of parabolics. It isalso well known that proofs in the case with parabolics are much morecumbersome and technical but follow the same arguments as the proofsin the purely hyperbolic case. For the sake of clarity and transparencyof exposition, we decided to prove Theorem 1.2 only in the absence ofparabolics. We state now the general results and discuss what changeshave to be made in the arguments given above.

Throughout this section we assume that the reader is familiar withbasic geometric facts about hyperbolic manifolds with cusps, [MT98].

As mentioned in the introduction, Evans [Eva04] obtained the follow-ing extension of Theorem 1.1.

Theorem 7.1 (Evans[Eva04]). Assume Γ is a finitely generated groupand that (ρi) is a sequence of faithful representations in D(Γ) convergingalgebraically to some representation ρ. If the convergence ρi → ρ isweakly type preserving, then ρi converges geometrically to ρ(Γ).

Recall that an algebraically convergent sequence ρi → ρ is weaklytype preserving if for every γ ∈ Γ with ρ(γ) parabolic, there is iγ withρi(γ) parabolic for all i ≥ iγ.

Theorem 1.4 can be extended to the weakly type preserving settingas follows:

Theorem 7.2. Let Γ be a finitely generated group, and (ρi) a sequencein D(Γ) converging algebraically to a representation ρ and geometricallyto a subgroup G of PSL2 C. If

• the convergence ρi → ρ is weakly type preserving, and• maximal cyclic subgroups of ρ(Γ) are maximally cyclic in G,

then G = ρ(Γ).

35

To begin with, set Mi = H3/ρi(Γ), MA = H3/ρ(Γ), MG = H3/G andlet π : MA → MG be the covering induced by the inclusion ρ(Γ) ⊂ G.We first recall the basic idea of the proof in the case without cusps.Since maximal cyclic subgroups of ρ(Γ) are maximally cyclic in G, thereare arbitrarily large standard compact cores C of MA such that thecomponents of ∂C facing convex cocompact ends of MA are strictlyconvex and the components facing degenerate ends embed under thecovering π : MA →MG. Composing the restriction π|C with the almostisometric embeddings π(C) → Mi supplied by geometric convergence,we obtain maps fi : C →Mi such that

(1) if a component S of ∂C faces a degenerate end of MA then fi|Sis an embedding for all large i,

(2) if S ⊂ ∂C faces a convex cocompact end then fi|S is a conveximmersion.

We then construct for large i a 3-manifold Ni containing C and a cov-ering fi : Ni → Mi with fi|C = fi. This covering is π1-surjective andhence a diffeomorphism, so in particular C embeds under the coveringπ : MA → MG. Since C can be chosen to be arbitrarily large, thisproves that the covering π is trivial and hence G = ρ(Γ).

If there are parabolics in the algebraic limit, the ends of MA are morecomplicated and refinements of the tools above are needed. The naturalreplacement of the compact core C is a submanifold C ⊂MA with thefollowing properties:

(1) if a component S of ∂C faces a degenerate NP-end of MA thenS embeds under the covering MA →MG,

(2) if S faces a geometrically finite NP-end then S is strictly convex,(3) the complement in C of the µ-cuspidal part M cusp<µ

A of MA is astandard compact core, where µ is the Margulis constant.

The construction of such a submanifold is the same as that used toproduce the compact cores above: one takes a large metric neighbor-hood of the convex core CC(MA) and deletes standard neighborhoodsof the degenerate NP-ends of MA. In this case, however, the resultingmanifold C will contain parts of the cusps of MA, and will thereforebe noncompact. This is a problem, since π(C) ⊂ MG will also be non-compact and the almost isometric embeddings from MG to Mi providedby geometric convergence are only defined on compact sets. However,we still can use the almost isometric embeddings to produce locallybilipschitz maps fi : C \M cusp<µ

A → Mi, and we will show that if theconvergence ρi → ρ is weakly type preserving then these can be ex-tended to locally bilipschitz maps

fi : C →Mi


converging uniformly on compact sets to the restriction π|C . The proofof Theorem 7.2 is then word-by-word the same as the proof Theorem1.4 with the maps fi playing the role of fi.

It remains to construct fi. Consider the maps

fi : C \M cusp<µA →Mi

described above, and let ε be a small positive constant. If i is largethen fi is locally (1 + ε)-bilipschitz; combined with the fact that theconvergence ρi → ρ is weakly type preserving, this implies that fi sendsloops homotopic into the cusps of MA to parabolic loops in Mi withnearly the same length. It follows that C ∩ ∂M cusp<µ

A is mapped underfi into a small neighborhood of ∂M cusp<µ

i . After a small perturbation,we can then arrange that

(1) fi(C ∩ ∂M cusp<µA ) ⊂ ∂M cusp<µ

i ,(2) Dfi sends vectors orthogonal to ∂M cusp<µ

A to vectors orthogonalto ∂M cusp<µ

i .

Brock and Bromberg [BB04, Lemma 6.16] show how to accomplish sucha perturbation with a bit of finesse; in particular, their argument showsthat we can still assume that fi is locally bilipschitz.

Recall that C was constructed by removing standard neighborhoodsof the degenerate NP-ends of MA from its convex core. These neighbor-hoods can be chosen so that their intersections withM cusp<µ

A are foliatedby geodesic rays orthogonal to ∂M cusp<µ

A ; the intersection C ∩M cusp<µA

then enjoys the same property. Combining this with (1) and (2) above

allows us to extend fi to a locally bilipschitz map fi : C → Mi as fol-lows: define fi to coincide with fi on C \M cusp<µ

A and map geodesicrays in C ∩M cusp<µ

A orthogonal to ∂M cusp<µA isometrically to geodesic

rays orthogonal to ∂M cusp<µi . A quick computation in the upper half

space model for H3 verifies that the extension fi is also locally (1 + ε)-

bilipschitz. It follows that fi → π|C uniformly on compact subsets.

Observe that as in the case without parabolics, we can replace thesecond hypothesis of Theorem 7.2 with the condition that every degen-erate end of MA embeds in MG.

A version of Proposition 5.1 to the setting with cusps is also readilyestablished:

Proposition 7.3. Let M and N be hyperbolic 3-manifolds with infinitevolume and let τ : M → N be a covering. Assume that π1(M) is finitelygenerated and that M has a degenerate end which does not embed underthe covering τ . Then there is a hyperbolic 3-manifold M ′ with finitely

37

generated fundamental group, with

3χ(M ′) + #cusps in M ′ > 3χ(M) + #cusps in Mand coverings τ ′ : M →M ′ and τ ′′ : M ′ → N with τ = τ ′′ τ ′.

The proof is the same as that of Proposition 5.1, except that insteadof considering the deformation space CH(M) of convex-cocompact hy-perbolic structures on M one uses geometrically finite metrics whoseparabolic loci coincide with that of the original hyperbolic structure onM . The space of such metrics, up to isometries isotopic to the identitymap, is a complex manifold of C-dimension

−3χ(M)−#cusps in M ≥ 0.

After these remark the proof of Proposition 7.3 is the same as the proofof Proposition 5.1.

Having provided versions of Theorem 1.4 and Proposition 5.1 thatapply to representations with parabolics, we are almost ready to discussthe general form of Theorem 1.2. Before doing so, we need a definition:

Definition. Assume that a sequence of subgroups (Gi) of PSL2 C con-verges geometrically to a subgroup G. We say that the convergenceGi → G is geometrically weakly type preserving if for every g ∈ Gparabolic there is a sequence (gi) with gi ∈ Gi converging to g and withgi parabolic for all but finitely many i.

In the terminology of [BS06], a geometrically convergent sequence ofsubgroups converges in a geometrically weakly type preserving mannerif and only if the associated sequence of pointed manifolds has uniformlength decay.

The general version of Theorem 1.2 reads now:

Theorem 7.4. Let Γ be a finitely generated group and (ρi) a sequence inD(Γ) that converges algebraically to a representation ρ and geometricallyto a subgroup G of PSL2 C. If the convergence ρi → G is geometricallyweakly type-preserving, then G is finitely generated.

As in the proof of Theorem 1.2 we choose a finitely generated sub-groupH ⊂ G containing ρ(Γ) and maximizing the quantity 3χ(H3/H)+2#cusps in H3/H. We claim that G = H. As before, there is a se-quence of representations σi : H → PSL2 C converging algebraically tothe inclusion of H in G and with σi(H) = ρi(Γi) for all i. In partic-ular, G is the geometric limit of the groups σi(H). The assumptionthat the convergence σi(H) = ρi(Γ) → G is geometrically weakly type-preserving implies that the algebraic convergence of σi to the inclusionof H into G is (algebraically) weakly type preserving. In particular


we deduce from Theorem 7.2 that either G = H or a degenerate endof H3/H does not embed under the cover H3/H → H3/G. The sec-ond possibility is ruled out by the choice of H and Proposition 7.3, soG = H and therefore is finitely generated. This concludes the proof ofTheorem 7.4.

Appendix A. Some assorted results

As mentioned in the introduction, we discuss here to which extentsome other results concerning faithful representations remain true if thecondition of faithfulness is relaxed.

Definition. A sequence (ρi)i of representations is eventually faithful iffor all γ ∈ Γ there is some iγ such that γ /∈ Ker(ρi) for all i ≥ iγ.

In some sense, every algebraically convergent sequence is eventallyfaithful. The following Lemma formalizes this; its proof is a simpleapplication of the Margulis Lemma.

Lemma A.1. Let Γ be a finitely generated group and (ρi) a sequencein D(Γ) converging algebraically to a representation ρ. Consider the

quotient group Γ = Γ/Ker(ρ) and let π : Γ → Γ be the associated pro-jection. Then there is an eventually faithful sequence of representationsσi : Γ → PSL2 C converging algebraically to a representation σ suchthat ρi = σi π and ρ = σ π.

We assume from now on that (ρi) is an eventually faithful sequenceof representations in D(Γ). If S ⊂ Γ is a finite generating set, then eachrepresentation ρi determines a convex function

dρi: H3 → R, dρi

(x) =∑γ∈S

dH3(x, ρi(γ)x).

Conjugating our representations by PSL2 C if necessary, we may assumethat some base point 0 ∈ H3 is the unique minimum of each dρi

. In thiscase we say that the sequence (ρi) consists of normalized representationsand we set dρi

= dρi(0). It is well-known that the sequence (ρi) contains

an algebraically convergent subsequence if

lim inf dρi<∞.

Otherwise, the sequence of actions of Γ via ρi on the scaled hyperbolicspaces 1

dρiH3 contains a subsequence which converges in the equivari-

ant Gromov-Hausdorff topology to a non-trivial action Γ y T on somereal tree T . Recall that an action on a tree is called trivial if it hasglobal fixed points. Morgan-Shalen [MS84], Paulin [Pau88] and Bestv-ina [Bes88] proved that if the representations ρi are faithful the action

39

Γ y T is small, meaning that the stabilizers of non-degenerate seg-ments in T are virtually abelian. Their arguments still apply if thesequence is only eventually faithful, see [NS]:

Theorem A.2. Every eventually faithful sequence of normalized repre-sentations in D(Γ) has a subsequence that either converges algebraicallyin D(Γ) or converges in the Gromov Hausdorff topology to a nontrivialsmall action Γ y T on a R-tree T .

It is theorem of Morgan-Shalen [MS84] that if the fundamental groupof a compact, irreducible and atoroidal 3-manifold M admits a nontriv-ial small action on a real tree, then either ∂M is compressible or thereare essential properly embedded annuli in (M,∂M). Combining thisfact with Theorem A.2 we obtain the following result, essentially dueto Thurston:

Corollary A.3. Assume that Γ is the fundamental group of a compact3-manifold with incompressible and acylindrical boundary. Then everyeventually faithful sequence of normalized representations in D(Γ) con-tains a convergent subsequence.

Many other results, for instance Thurston’s double limit theorem[Ota96] or the results in [KS02], ensuring the existence of convergentsubsequences of sequences of faithful representations can be reducedto the non-existence of certain actions of groups on trees; all theseresults still hold with obvious variations in the statements for eventuallyfaithful sequences. It may be therefore surprising that some convergenceresults for sequences of faithful representations completely fail in ourmore general setting.

In [Thu86A], Thurston proved a generalization of Corollary A.3 to thecase that Γ is the fundamental group of a compact 3-manifold M withincompressible boundary. More precisely, the so-called only-windows-break theorem asserts that whenever (ρi) is a sequence of faithful repre-sentations of Γ = π1(M) and N ⊂M is a component of the complementof the characteristic manifold, then the sequence (ρi|π1(N)) has, up toconjugacy, a convergent subsequence. Leaving the interested reader toconsult [Jac80] for information about the characteristic manifold, welimit ourselves to the following concrete example.

Let H be a handlebody of genus 2 and γ ⊂ ∂H a simple closedcurve on the boundary of H such that ∂H \ γ is incompressible andacylindrical; for instance, γ can be taken in the Masur domain of H[KS02]. We consider the manifold N obtained by doubling H alongN (γ), where N (γ) is a regular neighborhood of γ in ∂H. The manifoldN has incompressible boundary and there is, up to isotopy, a single


properly embedded essential annulus A ⊂ N , the annulus along whichwe have glued. The annulus A cuts N open into two copies H1 andH2 of H. In this particular example, Thurston’s only-windows-breaktheorem asserts:

Theorem A.4 (Thurston). Let N be as above and (ρi) a sequence ofdiscrete and faithful representations of π1(N) into PSL2 C. Then thesequence of restrictions

ρi|π1(H1) : π1(H1) → PSL2 Chas a subsequence that converges up to conjugacy in PSL2 C.

We claim that there is an eventually faithful sequence of discreterepresentations of π1(N) for which the claim of Theorem A.4 fails.

By construction, the manifold N is the double of H along N (γ). Let

τ : N → H

be that map given by ”folding” along N (γ). Identifying H with H1,one of the two pieces of N , we choose a base point p ∈ H and hence wehave a homomorphism

τ∗ : π1(N, p) → π1(H, p)

We consider also the Dehn-twist

δ : N → N

along the annulus A. It is easy to see that the sequence

τ∗ δn∗ : π1(N, p) → π1(H, p)

is eventually faithful, meaning that every element is contained in thekernel of τ∗ δn

∗ for finitely many n. By construction, τ∗ δn∗ is the

identity on π1(H, p) for all n.

Remark. Essentially we are saying that π1(N) is fully residually free.This fact holds for every group obtained by doubling a free group alonga cyclic subgroup.

Since H is a handlebody, π1(H, p) is a free group. Hence, we canchoose some sequence of faithful representations (σn) ∈ D(π1(H)) whichcannot be conjugated to obtain convergent subsequences. Setting ρn =σn τ∗ δn

∗ we obtain an eventually faithful representation which doesnot contain any subsequence whose restriction to π1(H) converges upto conjugacy.

Theorem A.5. The only-windows-break theorem fails for eventuallyfaithful sequences of representations.

41

An alternative statement of Theorem A.5 could be that when one isnot absolutely faithful more than the windows can get broken.

Appendix B. An independent remark

Recall that a Heegaard splitting of a closed 3-manifold M is a decom-position of M into two handlebodies, and the genus of these handlebod-ies is called the genus of the splitting. The Heegaard genus g(M) of Mis then the smallest possible g for which M admits a genus g Heegaardsplitting. It is easy to see that the fundamental group of M can begenerated by g(M) elements. In other words,

rank(π1(M)) ≤ g(M)

where the rank of a group is the smallest number of elements needed togenerate it.

In [BZ84], Boileau-Zieschang constructed examples of 3-manifoldswith rank(π1(M)) < g(M); other examples have been constructed bySchultens and Weidmann [SW], but there are no known closed hyper-bolic 3-manifolds with this property.

Motivated by the examples presented in this paper, we observe:

Proposition B.1. Assume that there is a finite volume hyperbolic 3-manifold M with a single cusp and a degree 2 cover M ′ → M , whereM ′ has again a single cusp and rank(π1(M

′)) < rank(π1(M)). Thenthere are closed hyperbolic 3-manifolds N with rank(π1(N)) < g(N).

Alan Reid and Hyam Rubinstein have given constructions of closedhyperbolic 3-manifolds which have degree 2 covers with smaller rank.Unfortunately, the authors of this note were unable to modify theirconstruction to find a manifold as needed in Proposition B.1.

Proof. Let M and M ′ be the compact 3-manifolds whose interiors arehomeomorphic to M and M ′ respectively. Extend the cover M ′ →M toa cover τ : M ′ → M and choose p′ ∈ ∂M ′ and p ∈ ∂M with τ(p′) = p.

The condition that M ′ has a single cusp implies that the groupτ∗(π1(∂M

′, p′)) has index 2 in π1(∂M, p). In particular, the latter isgenerated by τ∗(π1(M

′, p′)) and any element that lies outside of it.Choose a primitive element γ ∈ π1(∂M, p) that does not belong toτ∗(π1(∂M

′, p′)), and let Mγ be the 3-manifold obtained by Dehn-fillingM along γ. Then there is a surjective homomorphism π1(M) → π1(Mγ);since γ is in its kernel, the subgroup τ∗(π1(M

′, p′)) ⊂ π1(M) maps ontoπ1(Mγ). In particular, we have

rank(π1(Mγ)) ≤ rank(π1(M′)) < rank(π1(M))


On the other hand, it follows from Thurston’s Dehn-filling theorem[BP92] and the work of Moriah-Rubinstein [MR97] that if γ is outsideof a finite collection of lines in Z2 = π1(∂M, p) then Mγ is hyperbolicand g(Mγ) = g(M). Here, the Heegaard genus g(M) is defined as itis for closed 3-manifold, except that we consider decompositions of Minto compression bodies rather than handlebodies. The relationshipbetween rank and Heegaard genus in this extended setting is a little bitmore complicated, but since M has a single cusp one can check thatagain rank(π1(M)) ≤ g(M). Therefore,

rank(π1(Mγ)) < rank(π1(M)) ≤ g(M) = g(Mγ)

as desired.

References

[Ago] I. Agol, Tameness of hyperbolic 3-manifolds, arXiv:math.GT/0405568[AC96A] J. Anderson and R. Canary, Algebraic limits of Kleinian groups which

rearrange the pages of a book, Invent. Math. 126 (1996), no. 2, 205–214.[AC96B] J. Anderson and R. Canary, Cores of hyperbolic 3-manifolds and limits

of Kleinian groups. Amer. J. Math. 118 (1996).[AC00] J. Anderson and R. Canary, Cores of hyperbolic 3-manifolds and limits

of Kleinian groups. II, J. London Math. Soc. (2) 61 (2000).[BP92] R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Springer-

Verlag, (1992).[Bes88] M. Bestvina, Degenerations of the hyperbolic space, Duke Math. J. (1988).[BZ84] M. Boileau and H. Zieschang, Heegaard genus of closed orientable Seifert

3-manifolds Invent. Math. 76 (1984).[Bon86] F. Bonahon, Bouts des varietes hyperboliques de dimension 3, Ann. of

Math. (2) 124 (1986), no. 1, 71–158.[BO88] F. Bonahon and J.-P. Otal, Varietes hyperboliques a geodesiques arbi-

trairement courtes, Bull. London Math. Soc. 20 (1988), 255-261.[Bro01] J. Brock, Iteration of mapping classes and limits of hyperbolic 3-

manifolds, Invent. Math. 143 (2001), no. 3, 523–570.[Bro01] J. Brock, Continuity of Thurston’s length function , Geom. Funct. Anal.

10 (2000), no. 4, 741–797.[BB04] J. Brock and K. Bromberg, On the density of geometrically finite Kleinian

groups, Acta Math. 192, (2004), 33-93.[BS06] J. Brock and J. Souto, Algebraic limits of geometrically finite manifolds

are tame, Geom. Funct. Anal. 16 (2006), no. 1, 1–39.[CG06] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic

3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385–446[Can91] R. D. Canary, The Poincare metric and a conformal version of a theorem

of Thurston , Duke Math. J. 64 (1991), no. 2, 349u359.[Can93] R. D. Canary, Ends of hyperbolic 3-manifolds, J. Amer. Math. Soc. 6

(1993), no. 1, 1–35.[Can96] R. D. Canary, A covering theorem for hyperbolic 3-manifolds and its

applications, Topology 35 (1996).

43

[CCHS] D. Canary, M. Culler, S. Hersonsky and P. Shalen, Free Kleinian groupsand volumes of hyperbolic 3-manifolds.

[Com96] T. Comar, Hyperbolic Dehn surgery and convergence of Kleinian groups,PhD thesis, University of Michigan (1996).

[Eva04] R. Evans, Weakly type-preserving sequences and strong convergence,Geom. Dedicata 108 (2004).

[HP04] M. Heusener and J. Porti, The variety of characters in PSL2(C), Bol.Soc. Mat. Mexicana (3) 10 (2004), Special Issue, 221–237.

[Jor73] T. Jorgensen, On cyclic groups of Mobius transformations , Math. Scand.33 (1973), 250–260.

[Jor76] T. Jorgensen, On discrete groups of Mobius transformations , Amer. J.Math. 98 (1976), no. 3, 739–749.

[Jac80] Jaco, Lectures on three-manifold topology, CBMS Regional ConferenceSeries in Mathematics, 43. American Mathematical Society, 1980.

[Kap01] M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Math-ematics, 183. Birkheuser 2001.

[KT90] S. Kerckhoff and W. Thurston, Noncontinuity of the action of the modulargroup at Bers’ boundary of Teichmfller space, Invent. Math. 100 (1990),no. 1, 25–47.

[Kle05] G. Kleineidam, Strong convergence of Kleinian groups, Geom. Funct.Anal. 15 (2005).

[KS02] G. Kleineidam and J. Souto, Algebraic convergence of function groups,Comment. Math. Helv. 77 (2002).

[Mar74] A. Marden, The geometry of finitely generated Kleinian groups, Ann. ofMath. 99 (1974).

[MT98] K. Matsuzaki, M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups,Oxford Mathematical Monographs. Oxford Science Publications. TheClarendon Press, Oxford University Press, New York, (1998).

[Mc91] C. McMullen, Cusps are dense, Ann. of Math. 133 (1991), 217ð247.[Mc99] C. McMullen, Hausdorff dimension and conformal dynamics I. Strong

convergence of Kleinian groups, J. Differential Geom. 51 (1999), no. 3,471–515.

[MS84] J. Morgan and P. Shalen, Valuations, trees, and degenerations of hyper-bolic structures I, Ann. of Math. (2) 120 (1984), no. 3, 401–476.

[MR97] Y. Moriah and H. Rubinstein, Heegaard structures of negatively curved3-manifolds, Comm. Anal. Geom. 5 (1997).

[NS] H. Namazi and J. Souto Heegaard splittings and pseudo-Anosov maps,preprint 2006.

[Ota94] J.-P. Otal, Sur la degenerescence des groupes de Schottky, Duke Math. J.,74 (1994).

[Ota96] J.-P. Otal, Theoreme d’hyperbolisation pour les variete fibrees de dimen-sion 3, Asterisque, Societe Mathematique de France, (1996).

[Ota98] J.-P. Otal, Thurston’s hyperbolization of Haken manifolds, Surveys in dif-ferential geometry, Vol. III (Cambridge, MA, 1996), 77–194, Int. Press,Boston, MA, 1998.

[Pau88] F. Paulin, Topologie de Gromov equivariante, structures hyperboliques etarbres reels, Invent. Math. 94 (1988).

[SW] J. Schultens and R. Weidmann, On the geometric and algebraic rank ofgraph manifolds, preprint.


[Thu86A] W. P. Thurston, Hyperbolic structures on 3-manifolds I: Deformation ofacylindrical manifolds, Annals of Math. 124 (1986).

[Thu86B] W. P. Thurston, Hyperbolic structures on 3-manifolds II: Surface groupsand 3-manifolds which fiber over the circle, math.GT/9801045

Ian Biringer, Department of Mathematics, University of [email protected]

Juan Souto, Department of Mathematics, University of Michigan,[email protected]

Documents

Introduction - Yale Universityusers.math.yale.edu/~ib93/limits.pdf · Introduction When Γ is a ﬁnitely generated group, let D(Γ) be the set of all representations ρ: Γ → PSL