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Fundamental Groups and Limits of Almost

Homogeneous Spaces.

Sergio Zamorasxz38@psu.edu

Abstract

We say that the sequence of groups Γn acts almost transitively on asequence of proper metric spacesXn if for every n, there is an isometricdiscrete cocompact action of Γn on Xn such that the diameters of thequotients Xn/Γn converge to 0 as n → ∞.

In such a case, we prove that if the sequence Xn consists of lengthspaces and converges in the pointed Gromov–Hausdorff sense to aproper length space X, then X is isometric to a nilpotent locallycompact group equipped with an invariant length metric.

Furthermore, assuming X is either finite dimensional or semilo-cally simply connected, we show that it is a Lie group equipped witha Finsler or sub-Finsler metric, and for large enough n, there are sub-groups Λn ≤ π1(Xn) with surjective morphisms Λn → π1(X).

1 Introduction.

Metric spaces with groups of isometries acting transitively have been studiedextensively (c.f. [4], [5], [6], [20], [22]). In here we will focus on the onesobtained as Gromov–Hausdorff limits of proper length spaces with isometrygroups acting discretely and almost transitively.

Definition 1. Let X be a metric space and Γ a group acting on X byisometries. We say that Γ acts discretely if for all x ∈ X , r > 0, the set

γ ∈ Γ|d(γx, x) < r

is finite.

Definition 2. Let ε > 0, X a metric space, and a group Γ acting on Xby isometries. We say that Γ acts ε-transitively on X if for any x, y ∈ X ,there is g ∈ Γ such that d(gx, y) < ε. Equivalently, an isometric action isε-transitive if every orbit intersects every open ball of radius ε.

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Definition 3. We say that a sequence of groups Γn acts almost transitively

on a sequence of metric spaces Xn if there is a sequence εn ∈ R+ such thatfor every n, Γn acts isometrically, εn-transitively and discretely on Xn, andεn → 0 as n → ∞. In such a case, we say that the sequence Xn consists ofalmost homogeneous spaces.

Remark 4. A sequence of homogeneous length spaces Xn is in general nota sequence of almost homogeneous spaces because the group actions are notnecessarily discrete. So the sequences of almost homogeneous spaces do notgeneralize the concept of homogeneous spaces.

1.1 Main Problem.

Let (Xn, pn) be a sequence of proper almost homogeneous length spaces con-verging in the pointed Gromov–Hausdorff sense to a proper length space(X, p). Our goal is to classify the spaces X that can arise this way and studythe relationship between those spaces and the sequence Xn.

Example 5. This situation arises naturally when we take Xn to be a se-quence of Galois covers Xn → Yn of a sequence Yn of length spaces withdiam(Yn) → 0. A priori there is no need for the sequence Xn to have alimit, but in the particular case when the spaces Yn are closed Riemannianmanifolds of fixed dimension and Ric ≥ −1, Gromov’s compactness criterionguarantees the existence of a limit space X up to subsequence ([17], Chapter5).

In the case when the limit space X is a compact Lie group, Alan Turingshowed [28] that it is necessarily a torus. Using this result, Tsachik Gelanderstudied the case when X is compact.

Theorem 6. [12]. Let Xn be a sequence of compact almost homogeneouslength spaces converging in the Gromov–Hausdorff sense to a compact lengthspace X . Then X is a torus, i.e. homeomorphic to a finite or infinite productof circles.

Corollary 7. Let Xn be a sequence of compact almost homogeneous lengthspaces converging in the Gromov–Hausdorff sense to a space X homeomor-phic to a closed manifold. Then X is homeomorphic to a finite dimensionaltorus.

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When X is not compact, the situation is more flexible and allows moredegenerate behaviour. Itai Benjamini, Hilary Finucane and Romain Tesserahave worked on this problem using Pansu’s Theorem to obtain an asmptoticresult.

Definition 8. For a metric space Y and a point q ∈ Y , we say that thepointed metric space (Y0, q0) is a partial asymptotic cone of Y if there is asequence λk ∈ R+ with λk → 0 such that

(λkY, q) → (Y0, q0) in the pointed GH sense.

If the convergence (λkY, q) → (Y0, q0) holds for any sequence λk → 0, we saythat (Y0, q0) is the asymptotic cone of Y .

Theorem 9. (Pansu, [24]). Let G be a group and S a finite set of gener-ators. If G is of polynomial growth, then the asymptotic cone of the Cayleygraph asociated to (G, S) exists and it is a simply connected nilpotent Liegroup N equipped with an invariant sub-Finsler metric.

Theorem 10. [2]. Let (Xn, pn) be a sequence of proper almost homogeneouslength spaces converging in the pointed Gromov–Hausdorff sense to a properlength space (X, p). Then the asymptotic cone of X exists and it is a sim-ply connected nilpotent Lie group N equipped with an invariant sub-Finslermetric.

Although Theorem 10 is not explicitly stated in [2], the argument fortheir Theorem 3.2.2 also proves Theorem 10. In this note we show a similarresult classifying the possible limit spaces.

Theorem 11. Let (Xn, pn) be a sequence of proper almost homogeneouslength spaces converging in the pointed Gromov–Hausdorff sense to a properlength space (X, p). Then X is isometric to a locally compact nilpotent groupequipped with a left invariant metric.

Our main result concerns when X is semilocally simply connected.

Theorem 12. Let (Xn, pn) be a sequence of proper almost homogeneouslength spaces converging in the pointed Gromov–Hausdorff sense to a properlength space (X, p). If X is semilocally simply connected, then X = N/Λ,where N is a simply connected nilpotent Lie group equipped with an invariantFinsler or sub-Finsler metric and Λ ≤ N is a central discrete subgroup.

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For large enough n, there are subgroups Λn ≤ π1(Xn, pn) with surjectivemorphisms

Λn → Λ = π1(X, p). (1)

Remark 13. The hypothesis ofX being semilocally simply connected can bereplaced by X having finite topological dimension, because of the followingtheorem (solution to Hilbert’s fifth problem) by Deane Montgomery and LeoZippin.

Theorem 14. [22]. Let X be a homogeneous proper length space of fi-nite dimension, then X is homeomorphic to a topological manifold, and itsisometry group is a Lie group.

1.2 Lower Semicontinuity of π1.

In this section we focus our attention on Equation 1, a lower semicontinuityof the fundamental group. It was previously known to hold when X wascompact.

Theorem 15. ([17], Section 3E). Let Yn be a sequence of compact lengthspaces converging to a compact semilocally simply connected length spaceY . Then for large enough n, there are surjective morphisms

π1(Yn) → π1(Y ).

This property is further studied by Christina Sormani and Guofang Weiin [25], [26], [27]. This lower semicontinuity does not hold when X is notsemilocally simply connected, as one can see from the following example.

Example 16. Let Yk := 1kS1 be the re-scaled unit circle with its length

metric. Then the sequence

Xn :=∏n

k=1 Yk

converges in the GH sense to

X :=∏∞

k=1 Yk,

where the products are taken using the Pythagoras Theorem. On the otherhand, the fundamental groups satisfy

π1(Xn) = Zn, and π1(X) = ZN,

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so there is no surjective morphism π1(Xn) → π1(X) for any n.

Theorem 12 states that under a symmetry assumption, we can extendTheorem 15 to the case when the limit is not compact. The following exampleshows that the discreteness hypothesis in Theorem 12 is necessary.

Example 17. Let Y be S1 with its standard metric of length 2π and Zn be

S3 with the round (bi-invariant) metric of constant curvature 1/n. Let Xn

be the quotient (Y × Zn)/S1 where S1 acts on Y × Zn as follows:

z(w, q) = (wz−1, zq) : z, w ∈ S1, q ∈ S

3.

Then Xn is isometric to S3 equipped with a re-scaled Berger metric. The se-quence Xn consists of simply connected homogeneous spaces, but its pointedGromov–Hausdorff limit is S1 × R2, which is not simply connected.

1.3 Existence of the Limit.

One may also study the existence of a partial limit (X, p) given a sequenceof proper length spaces (Xn, pn) with groups Γn acting discretely and almosttransitively. If the spaces Xn satisfy a uniform Ricci curvature-dimensioncondition (they are all CD(K,N) spaces) the limit (X, p) exists up to sub-sequence, and by the Bishop–Gromov inequalty, X has finite dimension, soby Remark 13, Theorem 12 holds.

Corollary 18. Let Yn be a sequence of d-dimensional closed Riemannianmanifolds with Ric ≥ −1, and diam(Yn) → 0. Then, up to subsequence, thesequence Xn of universal covers, converges to a simply connected nilpontetLie group equipped with an invariant Riemannian metric.

Remark 19. Vitali Kapovitch, Burkhard Wilking, and independently Em-manuel Breuillard, Ben Green, Terence Tao showed ([19] and [7], respec-tively) that under the hypothesis of Corollary 18, the limit is a nilpotent Liegroup equipped with an invariant Riemannian metric, so the only new partof Corollary 18 is the simple connectedness of the limit.

Benjamini, Finucane and Tessera also found a sufficient condition for thispartial limit to exist when the spaces Xn are graphs.

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Theorem 20. [2]. Let Dn ≤ ∆n be two sequences going to infnity, and letXn be a sequence of graphs. Assume there are groups Γn acting discretelyby isomorphisms and transitively on the vertices of Xn. If the balls of radiusDn satisfy

|B(x,Dn)| = O(Dqn)

for some q > 0, then (Xn, d/∆n) has a subsequence converging in the pointedGH sense to a nilpotent Lie group equipped with a left invariant sub-Finslermetric.

It is not known whether Theorem 20 holds if one removes the assumptionthat the groups Γn act discretely.

1.4 Summary.

All subsequent sections are focused on proving Theorems 11 and 12. Sections3 to 8 contain the proof of Theorem 12, assuming the limit space X , ishomeomorphic to a topological manifold (i.e. Theorem 56), while Section 9reduces the proof of Theorem 12 to that case (i.e. Theorem 95) and provesTheorem 11 in the process.

We will exploit the ultrafilter techniques from [7] and a holonomy mapsimiliar to the one in [8], [16].

In Section 2, we introduce our notation and the standard theory we willuse. Throughout the proof we will use the ultrafilter language. However, inthis outline we will say “for large enough n” (FLEn) for simplicity.

In Section 3, by repeated applications of the Generalized Margulis Lemmaof Breuillard, Green, and Tao, together with a local uniform doubling con-dition on X , we obtain FLEn that the groups Γn have subgroups Γ′

n withbounded index, having normal subgrops H ′

n ⊳ Γ′n with the property that the

H ′n orbit of pn is small, and Γ′

n/H′n is nilpotent with nilpotency step bounded

by a number independent of n. We replace Γn by Γ′n and restart the proof.

In Section 4, we show that X is a nilpotent Lie group with an invaraintFinsler or sub-Finsler metric, and FLEn, the groups Γn act in an almosttranslational way in arbitrarily large neighborhoods of pn.

In Section 5, we make use of the escape norm from [7] to find FLEn,normal subgroups Hn ⊳ Γn such that the Hn orbit of pn is small, so Xn andXn/Hn are GH close, and Zn := Γn/Hn contain large subsets Yn withoutnontrivial subgroups.

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This allows us to, in Section 6, use the space X as a model (see [7], Section3) for the ultralimit Y := lim

n→αYn. This implies that FLEn, there are large

nice subsets Pn of Zn (precisely, nilprogressions in C-regular form).In Section 7, we use the Malcev Embedding Theorem to find FLEn,

groups ΓPn isomorphic to lattices in simply connected Lie groups Gn, withisometric actions

Φn : ΓPn → Iso(Xn/Hn).

We notice, using an elementary result in algebraic topology, that the kernelsof those actions are isomorphic to quotients of π1(Xn, pn).

Finally, in section 8, we find, FLEn, subgroups of Ker(Φn) isomorphic toΛ = π1(X, p), all of them isomorphic to central discrete subgroups of simplyconnected nilpotent Lie groups, finishing the proof of Theorem 56.

On the other hand, in section 9 we specialize the theory of homogeneousproper length spaces developed by Valerii Berestovskii to our setting, showingTheorems 95 and 11.

The author would like to thank Vladimir Finkelshtein, Enrico LeDonne,Adriana Ortiz, Anton Petrunin and Burkhard Wilking for helpful commentsand discussions. This research was supported in part by the InternationalCentre for Theoretical Sciences (ICTS) during a visit for participating in theprogram - Probabilistic Methods in Negative Curvature (Code: ICTS/pmnc2019/03).

2 Preliminaries.

2.1 Notation.

For any element g in a group G, we will denote by Lg the left shift G → Ggiven by Lg(h) = gh. If G is abelian, we may denote Lg by +g. We say that aset A ⊂ G is symmetric if A = A−1 and e ∈ A. For subsets A1, . . . , Ak ⊂ G,we will denote by A1 . . . Ak the set of all products

a1 . . . ak|ai ∈ Ai ⊂ G.

We will denote by A1 × . . .×Ak the set of all sequences

(a1, . . . , ak)|ai ∈ Ai ⊂ Gk.

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If Ai = A for i = 1, . . . , k, we will also denote A1 . . . Ak by Ak, and A1× . . .×

Ak by A×k.

For H,K subgroups of a group G, we define the commutator subgroup

[H,K] to be the group generated by the elements [h, k] := h−1k−1hk withh ∈ H, k ∈ K. Define G(0) as G, and G(j+1) inductively as G(j+1) := [G(j), G].If G(s) = e for some s ∈ N, we say that G is nilpotent of step ≤ s.

For curves β, γ : [0, 1] → Y , we denote by β : [0, 1] → Y the curve givenby β(t) = β(1 − t). And if β(0) = γ(1), we denote by β ∗ γ : [0, 1] → Y theconcatenation

β ∗ γ(t) =

γ(2t) if t ≤ 1/2

β(2t− 1) if t ≥ 1/2.

If β(0) 6= γ(1), we say that β ∗γ is undefined. We call β ∗γ the concatenationof β and γ. We will write β ≃ γ whenever β and γ are homotopic relative totheir endpoints.

If Y is a length space, we say that a curve is an ε-lasso nailed at β(0) if itis of the form β ∗ γ ∗ β, with β(1) = γ(0), and γ a loop contained in a closedball of radius ε. For a pointed space (Y, y), we denote its loop space as

Ω(Y, y) := γ : [0, 1] → Y continuous |γ(0) = γ(1) = y.

2.2 Uniform Distance.

In a metric space (Y, d), we will denote the closed ball of center q ∈ Y andradius r > 0 as

BYd (q, r) := y ∈ Y |d(y, q) ≤ r.

We will sometimes omit d or Y and write B(q, r) if the metric space we areconsidering is clear from the context.

Definition 21. Let A,B be metric spaces and f, h : A → B. For a subsetC ⊂ A, we define the uniform distance between f and h in C as

dU(f, h, C) := supc∈C

d(f(c), h(c)).

Definition 22. Let A,B be two metric spaces and f : A → B a function.We define the distortion of f as

Dis(f) := supa1,a2∈A

|d(f(a1), f(a2))− d(a1, a2)|.

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The following proposition follows from the triangle inequality in the cor-responding spaces.

Proposition 23. Let f, g, h : A→ B1, f1, g1 : B1 → B2, and C ⊂ A. Then

dU(f, g, C) ≤ dU(f, h, C) + dU(h, g, C),dU(f1f, g1g, C) ≤ dU(f, g, C) + dU(f1, g1, g(C)) +Dis(f1|f(C)∪g(C)).

2.3 Ultrafilters.

In this section we discuss the ultrafilter tools we will use during the proofof Theorem 12, including metric ultralimits and algebraic ultraproducts. Werefer the reader to ([1], Chapter 4) , ([2], Section 2.1), ([7], Appendix A) forproofs and further discussions.

Definition 24. Let ℘(N) denote the power set of the natural numbers andα : ℘(N) → 0, 1 be a function. We say that α is a non-principal ultrafilter

if it satisfies:

• α(N) = 1.

• α(A ∪B) = α(A) + α(B) for all disjoint A,B ⊂ N.

• α(F ) = 0 for all finite F ⊂ N.

Using Zorn’s Lemma it is not hard to show that nonprincipal ultrafiltersexist. We will choose one (α) and fix it for the rest of these notes. SetsA ⊂ N with α(A) = 1 are called α-large. For a property P : N → 0, 1, ifα(P−1(1)) = 1 we will often say “P holds for n sufficiently close to α”, or“P holds for n sufficiently α-large”.

Definition 25. Let An be a sequence of sets. In the product

A′ :=∞∏

n=1

An,

we say that two sequences an, a′n are α-equivalent if

α(n|an = a′n) = 1.

The set A′ modulo this equivalence relation is called the algebraic ultraproductof the sets An and is denoted by

Aα := limn→α

An.

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If An = R for each n, an element in Aα is called a non-standard real

number. For x = xn, y = yn nonstandard real numbers, we say thatx ≤ y if α(n ∈ N|xn ≤ yn) = 1.

Definition 26. Let xn be a sequence in a metric space X . We say that thesequence ultraconverges to a point x∞ if for every ε > 0,

α(n|d(xn, x∞) < ε) = 1.

The point x∞ is called the ultralimit of the sequence and we write xnα−→ x∞,

orlimn→α

xn = x∞.

Since α is fixed, the ultralimit of a sequence only depends on the topologyof X and not on the particular metric. It is easy to show that if a sequencexn has an ultralimit, then it is unique. Furthermore, if X is compact, thenany sequence in X ultraconverges.

Definition 27. Let (Xn, pn) be a sequence of pointed metric spaces. Let X ′α

be the set of sequences xn ∈ Xn such that

supn∈N

d(xn, pn) <∞.

Equip X ′α with the pseudometric

d(xn, yn) := limn→α

d(xn, yn).

Let Xα be the metric space corresponding to the pseudometric space X ′α.

The pointed metric space (Xα, pα), where pα is the class of the sequence pn,is called the metric ultralimit of the sequence (Xn, pn). It is straightforwardto show that Xα is always a complete metric space.

Remark 28. If a sequence of proper metric spaces (Xn, pn) converges in thepointed GH sense to a proper metric space (X, p), then (Xα, pα) and (X, p)are isometric. Conversely, if the sequence (Xn, pn) is precompact in thepointed GH topology, then there is a subsequence that converges to (Xα, pα)in the pointed GH sense.

Remark 29. To define an algebraic ultraproduct Aα = limn→αAn or ametric ultraproduct Xα = limn→αXn, we don’t require the sets An or Xn tobe defined for every n, but only for all n in an α-large set.

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2.4 Approximate Isometries, Equivariant Convergence.

Definition 30. We say that f : A → B is an ε-approximation between Aand a subset C ⊂ B if Dis(f) < ε, and the Hausdorff distance between f(A)and C is less than ε.

Lemma 31. Let (Y, q) be a proper length space and φn : Y → Y a sequenceof maps satisfying

supn→∞

d(φnq, q) <∞.

Assume that for every R > 0, Dis(φn|B(q,R)) → 0, and consider the mapφα : Y → Y given by

φα(y) := limn→α

φn(y).

Then φα is an isometry, and for all R > 0,

limn→α

dU(φn, φα, B(q, R)) = 0.

We will call φα the ultralimit of the sequence φn.If we further assume that for every R, φn|B(q,R) are δn-approximations

between B(q, R) and B(φn(q), R) for numbers δnα−→ 0, then φα is surjective.

Definition 32. Let (Xn, pn), (Yn, qn) be two sequences of pointed metricspaces and let φn, ψn : Xn → Yn be two sequences of maps. We say that φnis ultraequivalent to ψn if for every R > 0,

limn→α

dU(φn, ψn, B(pn, R)) = 0.

Let (Xn, pn) be a sequence of pointed proper metric spaces, converging inthe pointed GH sense to a proper length space (X, p). By definition of GHconvergence, there are maps fn : Xn → X and hn : X → Xn with fn(pn) = pand the property that for all ε, R > 0, there is M ∈ N so that for all n > M ,there exists r ≥ R such that fn|B(pn,r) is an ε-approximation between B(pn, r)and B(p, r), and

dU(hn fn, id, B(pn, R)) < ε.

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Let Γn act by isometries on Xn. We say that a sequence gn ∈ Γn ultra-

converges to a map g : X → X if the sequence of maps (fn gn hn) isultraequivalent to the constant sequence g.

We say that a sequence gn ∈ Γn is stable if

supn∈N

d(gn(pn), pn) <∞.

The set Γα of stable sequences modulo ultraequivalence is called the equiv-

ariant ultralimit of the sequence Γn.

Lemma 33. The map Φ : Γα → Iso(X) that sends a sequence to its ul-tralimit is well defined (doesn’t depend on the representative in the equiva-lence class), and it is injective (that is, if two stable sequences have the sameequivariant ultralimit, then the sequences are ultraequivalent). Furthermore,if the sequence Γn acts almost transitively on the sequence Xn, then Γα actstransitively on X .

Remark 34. The set Γα has two equivalent group structures: The one ob-tained by pulling back the group structure in Iso(X) through Φ,

a ∗ b := Φ−1(Φ(a) Φ(b)).

And the one given by

gn ∗ g′n := gn g

′n.

Let (Y, q) be a proper pointed metric space. When we equip the isome-try group Iso(Y ) with the compact-open topology, the following family is acompact neighbourhood basis of the identity:

UYR,ε := g ∈ Iso(Y )| sup

y∈B(q,R)

d(g(y), y) < ε

with R, ε ∈ R. This gives Iso(Y ) the topology of a locally compact Hausdorffgroup.

Remark 35. If we equip Iso(X) with the topology described above, it iseasy to check that the image Φ(Γα) is a closed subgroup.

Remark 36. The topologies on Γα induced by the two following neighbor-hood bases of the identity are equivalent. The one obtained by the ultralimit:

UαR,ε := gn ∈ Γα|α(n|gn ∈ UXn

R,ε) = 1.

The topology inherited from Φ:

UαR,ε := Φ−1(UX

R,ε).

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2.5 Ultraconvergence of Polynomials.

Definition 37. Let Qn : Rℓ → Rm be a sequence of polynomials of boundeddegree. We say that the sequence converges well to a polynomial Q : Rℓ →Rm if the sequence Qn is ultraequivalent to the constant sequence Q. Equiv-alently, the sequence Qn converges well to Q if the sequences of coefficientsof Qn ultraconverge to the corresponding coefficients of Q.

Let [·, ·]n : Rm × Rm → R

m be a sequence of Lie algebra structures inRm. We say that the sequence converges well to a Lie algebra structure[·, ·] : Rm × Rm → Rm if for each i, j, k ∈ 1, . . . , m,

cki,j(n)α−→ cki,j,

where

[ei, ej]n =

m∑

k=1

cki,j(n)ek, [ei, ej] =

m∑

k=1

cki,jek.

Lemma 38. Let Qn : Rr × Rr → R

r be a sequence of polynomial groupstructures in Rr of bounded degree. Assume Qn converges well to a polyno-mial group structure Q : Rr × Rr → Rr. Then the corresponding sequenceof Lie algebra structures on Rr converges well to the Lie algebra structure ofQ.

Proof. This follows from the fact that the coefficients cki,j(n) depend contin-uously on the second derivatives of Qn, which by hypothesis, ultraconvergeto the corresponding second derivatives of Q.

Lemma 39. For each d ∈ N, there is N0 ∈ N such that the followingholds. Let IN0 := −1, . . . , −1

N0, 0, 1

N0, . . . , 1, and assume we have polynomials

Qn, Q : Rℓ → Rm of degree ≤ d such that Qn(x)α−→ Q(x) for all x ∈ (IN0)

×ℓ.Then Qn converges well to Q.

Proof. Working on each coordinate, we may assume that m = 1. We proceedby induction on ℓ, the case ℓ = 1 being elementary Lagrange interpolation.Name the variables x1, . . . , xℓ. Since

R[x1, . . . , xℓ] = (R[x1])[x2, . . . , xℓ],

we can consider the polynomials Qn, Q as polynomials Qn, Q in the variablesx2, . . . , xℓ with coefficients in R[x1].

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If Qn(x)α−→ Q(x) for all x ∈ (IN0)

×ℓ, we would have Qn(q, x′)

α−→ Q(q, x′)

for all q ∈ IN0 and x′ ∈ (IN0)×(ℓ−1). By the induction hypothesis, if N0 was

large enough, depending on d, the coefficients of Qn, which are polynomialsin R[x1], ultraconverge to the coefficients of Q whenever x1 ∈ IN0 . By thecase ℓ = 1, if N0 was large enough, the coefficients of Qn ultraconverge tothe coefficients of Q.

Definition 40. For x ∈ Rr, we define its support as

supp(x1, . . . , xr) := i ∈ 1, . . . , r|xi 6= 0.

For x, y ∈ Rr, we say that x y if i ≤ j for every i ∈ supp(x), j ∈ supp(y).We say that a polynomial group structure Q : Rr×Rr → Rr is quasilinear if

Q(x, y) = Q(x, 0) +Q(0, y) = x+ y

whenever x y.

Note that for any quasilinear group structure in Rr, the coordinate axesare one-parameter subgroups, and the exponential map

exp : T0Rr = R

r → Rr

is the identity when restricted to such axes.

Lemma 41. Consider quasilinear polynomial nilpotent group structuresQn, Q : Rr × Rr → Rr of bounded degree. Let logn, log : Rr → Rr = T0R

r

denote the logarithm maps for the group structures Qn and Q, respectively.Assume the sequence Qn converges well to Q, and a sequence xn ∈ R

r ultra-converges to a point x ∈ Rr. Then

limn→α

logn(xn) = log(x). (2)

Proof. Let xn = (xn,1, . . . , xn,r). By quasilinearity, we have

logn(xn) = logn (xn,1e1 + . . .+ xn,rer)

= logn ((xn,1e1) . . . (xn,rer))

= logn(expn(logn(xn,1e1)) . . . expn(logn(xn,rer)))

= logn(expn(xn,1e1) . . . expn(xn,rer))

By the Baker–Campbell–Hausdorff formula, the last expression is a polyno-mial in the variables xn,1, . . . , xn,r with coefficients depending continuouslyon the coefficients cki,j(n). By Lemma 38, Equation 2 is established.

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2.6 Constructing Covering Spaces.

Let ε > 0 and Γ be a group acting discretely ε-transitively by isometries ona proper length space Z.

Lemma 42. For any z ∈ Z, g ∈ Γ||gz − z| < 3ε generates Γ.

Proof. Let Γ′ be the subgroup generated by g ∈ Γ||gz − z| < 3ε andassume Γ′ 6= Γ. Let M = infd(gz, z)|g ∈ Γ\Γ′ and let g0 ∈ Γ\Γ′ suchthat d(g0z, z) = M . Take a minimizing geodesic c : [0,M ] → Z from g0zto z parametrized by arc length. By ε-transitivity, there is g1 ∈ Γ suchthat d(g1z, c(3ε/2)) < ε. It follows that d(g1z, z) < M , and g1 ∈ Γ′. Also,d(g−1

1 g0z, z) = d(g0z, g1z) < 3ε and g−11 g0 ∈ Γ′, contradicting that g0 ∈

Γ\Γ′.

Let z0 ∈ Z, ρ > 10ε, and set

B := B(z0, ρ),S := g ∈ Γ||gz0 − z0| ≤ 2ρ.

Let Γ be the abstract group generated by S, with relations

s = s1s2 in Γ, whenever s, s1, s2 ∈ S and s = s1s2 in Γ.

Then there is a canonical map Φ : Γ → Γ that extends the embeddingS → Γ. It is surjective by Lemma 42. Equip Γ with the discrete topology,and consider the topological space

Z := (Γ× B)/ ∼,

where ∼ is the minimal equivalence relation such that

(gs, x) ∼ (g,Φ(s)x) if s ∈ S, x,Φ(s)x ∈ B.

There is a well defined contnuous map Ψ : Z → Z given by

Ψ(g, x) := Φ(g)(x).

Definition 43. Let ε > 0. We say that a covering map Y → Y is ε-wide iffor every y ∈ Y , the ball BY (y, ε) is an evenly covered neighborhood of y.

Theorem 44. (Monodromy). The map Ψ is a regular ρ/3-wide coveringmap with Galois group Ker(Φ).

15

Proof Sketch: To prove Theorem 44, one has to observe that for every z ∈BZ(z0, ρ/2), every element z ∈ Ψ−1(z) has a unique representative of theform (g, z), with g ∈ Ker(Φ). From there, it is not hard to prove thatBZ(z0, ρ/2) is an evenly covered neighborhood of z0. Then the result followsfrom the fact that Ψ is compatible with the actions of Γ on Z and Z.

Lemma 45. Let Y be a length space, y ∈ Y , ε > 0, and ∆ a group. Thereis a regular ε-wide covering map Y → Y with Galois group ∆ if and only ifthere is a surjective map π1(Y, y) → ∆ whose kernel contains all the classescontaining ε-lassos nailed at y.

Lemma 45 follows from the standard construction of covering spaces (see[9] Chapter II, Sections VI-IX).

Theorem 46. Let A,B be proper length spaces with

dGH(A,B) ≤ ε/200,

and B → B a regular ε-wide covering map with Galois group ∆. Then thereis a regular covering map A→ A with Galois group ∆.

Proof. There is an ε/100-approximation f : A→ B. Let p ∈ A and q = f(a).Consider π1(B, q) → ∆ the map from Lemma 45 and let H be its Kernel.Construct a map

f♯ : Ω(A, p) → Ω(B, q)

as follows. For γ ∈ Ω(A, p), by absolute continuity, there is a partition

0 = t0 < . . . < tk = 1

such that γ([ti−1, ti]) is contained in a ball of radius ε/100 for each i ∈1, . . . , k. Let σi be a minimizing path from f(ti−1) to f(ti). Define f♯(γ) tobe the concatenation σk ∗ . . . ∗ σ1. Define ∧w∧ to be the minimal equivalencerelation in Ω(A, p) so that β∧w∧γ whenever

dU(β, γ, [0, 1]) ≤ε

100.

Also define ≡ to be the minimal equivalence relation in Ω(B, q) so that β ≡ γwhenever

dU(β, γ, [0, 1]) ≤ε

10.

One needs to verify that

16

• β ≃ γ ⇒ β∧w∧γ ⇒ f♯(β) ≡ f♯(γ) ⇒ ([f♯(β)])−1[f♯(γ)] ∈ H.

• f♯(β ∗ γ) ≡ f♯(β) ∗ f♯(γ).

• For every γ ∈ Ω(B, q), there is β ∈ Ω(A, p) such that f♯(β) ≡ γ.

• For any ε/2-lasso centered at p, its image under f♯ is ≡-equivalent toan ε-lasso centered at q.

Each item being an easy exercise in topology. We have a well defined surjec-tive map

f∗ : π1(A, p) → π1(B, q)/H = ∆

such that every class containing an ε/2-lasso is in the Kernel. By Lemma 45,there is a regular ε/2-wide covering map A→ A with Galois group ∆.

2.7 Lifting Curves.

Definition 47. We say that a continuous map Y → Z between path con-nected topological spaces has no content or has trivial content if the inducedmap π1(Y ) → π1(Z) is trivial. Otherwise, we say that the map has nontrivial

content.

Lemma 48. Let G be a connected nilpotent Lie group and K a compactsubgroup such that its connected component K0 is nontrivial. Then theinclusion K0 → G has nontrivial content.

Proof. Since K0 is a connected compact nilpotent Lie group, it is homeo-morphic to a torus, and π1(K0) is nontrivial. By the long exact homotopysequence of the fibration G→ G/K0, we have the exact sequence

π2(G/K0) → π1(K0) → π1(G)

Since compact subgroups of connected nilpotent Lie groups are central, G/K0

is a connected nilpotent Lie group, and is therefore aspherical (see [11], Sec-tion 1.2). This means that π2(G/K0) = 0, and the map π1(K0) → π1(G) isnontrivial.

Lemma 49. Let G, G be connected Lie groups such that G is a discreteextension of G (i.e. there is a surjective continuous morphism f : G → Gwith discrete kernel). Assume G and G are equipped with invariant length

17

metrics such that f is a local isometry. Let δ > 0 be such that BG(e, δ)contains no nontrivial subgroups, and the inclusion BG(e, δ) → G has no

content. Then BG(e, δ) contains no nontrivial subgroups and the inclusion

BG(e, δ) → G has no content.

Proof. If there is a group H ⊂ BG(p, δ), its image f(H) is a subgroup of Gcontained in BG(e, δ), so f(H) = e ≤ G. If there is a nontrivial elementh ∈ H\e, we can take a shortest path γ : [0, 1] → G from e to h. Theprojection f γ would be a noncontractible loop in G contained in BG(e, δ),contradicting the hypothesis that BG(e, δ) → G has no content. Also, wecan consider the commutative diagram

BG(e, δ) G

BG(e, δ) G

f

Since f is a covering map, the right vertical arrow induces an injectivemap at the level of fundamental groups, and the bottom horizontal arrowhas no content by hypothesis. Therefore the top horizontal arrow has trivialcontent as well.

Lemma 50. Let Y be a proper length space, and γ : [0, 1] → Y be acontinuous curve. Then for every ε > 0 there is a Lipschitz curve β : [0, 1] →Y with β(0) = γ(0), β(1) = γ(1), and

dU(γ, β, [0, 1]) ≤ ε.

Proof. Let k > 0 such that |x1 − x2| ≤ 3/k implies d(γ(x1), γ(x2)) ≤ ε/10for x1, x2 ∈ [0, 1]. Then choose β : [0, 1] → Y to be such that β|[j/k,(j+1)/k]

is a constant speed minimizing curve between γ(j/k) and γ((j + 1)/k) forj = 0, 1, . . . , k−1. It is easy to see that β satisfies the desired properties.

Lemma 51. Let f : Y → Z be a continuous map between proper lengthspaces. Assume that Y is semilocally simply connected, and the compositionB(y, r) → Y → Z has nontrivial content for some y ∈ Y , r > 0. Thenthere is a loop β : [0, 1] → Y based at y of length ≤ 3r such that f β isnoncontractible in Z.

18

Proof. Since Y is semilocally simply connected, there is ε > 0 such that anytwo closed curves β1, β2 : [0, 1] → B(y, 2r) with dU(β1, β2, [0, 1]) ≤ ε, arehomotopic to each other in Y . By hypothesis, there is a loop γ1 : [0, 1] →Y based at y whose image is contained in B(y, r) and such that f γ1 isnoncontractible in Z. By Lemma 50, there is a Lipschitz curve γ : [0, 1] →B(y, (1.1)r) such that f γ is noncontractible in Z. Let L be the Lipschitzconstant of γ, and pick ℓ ∈ N with ℓ ≥ 2L/r. For each j = 0, 1 . . . , ℓ, set σjto be a minimizing path from y to γ(j/ℓ), and define βj as the concatenationσj+1∗γ|[j/ℓ,(j+1)/ℓ]∗σj. Since γ is homotopic to the concatenation βℓ−1∗. . .∗β0,at least one of the curves βj is satisfies that f βj is noncontractible in Z,and all of them have length ≤ 3r.

Corollary 52. Let Y be a proper semilocally simply connected length space.Assume the inclusion B(y, r) → Y has nontrivial content for some y ∈ Y ,r > 0. Then there is a noncontractible loop based at y of length ≤ 3r.

Proof. Apply Lemma 51 with f = idY .

Definition 53. Let f : Y1 → Y2 be a map between proper length spaces.We say that f is a metric submersion if for every y ∈ Y1, and r > 0, we havef(BY1(y, r)) = BY2(f(y), r)

Lemma 54. Let Y be a proper length space, and Γ ≤ Iso(Y ) a closedsubgroup. Then the quotient map f : Y → Y/Γ is a metric submersion.

Proof. Let y ∈ Y , r > 0, and z ∈ Y/Γ with dY/Γ(f(y), z) ≤ r. Since Γ isclosed, the orbits are closed, and since Y is proper, there is z1 ∈ f−1(z) withd(y, z1) = d(f(y), z) ≤ r. This proves BY/Γ(f(y), r) ⊂ f(BY (y, r)). Theother contention is immediate from the definition of the metric in Y/Γ.

Lemma 55. Let f : Y1 → Y2 be a metric submersion between proper lengthspaces, γ : [0, 1] → Y2 a Lipschitz curve, and q0 ∈ f−1(γ(0)). Then there is acurve γ : [0, 1] → Y1 with f γ = γ, γ(0) = q0, and length(γ) = length(γ).

Proof. For each j ∈ N, let Dj := 0, 12j, . . . , 2

j−12j, 1, and define hj : Dj → Y1

as follows: Let hj(0) = q0, and inductively, let hj(x + 1/2j) be a point inf−1(γ(x+ 1/2j)), such that

dY1(hj(x+ 1/2j), hj(x)) = dY2(γ(x+ 1/2j), γ(x)), for x ∈ Dj\1.

Using Cantor’s diagonal argument, we can find a subsequence of hj that con-verges for every dyadic rational. Since the maps hj are uniformly Lipschitz,we can extend this map to a Lipschitz map γ : [0, 1] → Y1. It is easy to checkthat γ satisfies the desired properties.

19

3 Virtual Nilpotency of Γn.

As stated in the Summary, Sections 3 to 8 focus on proving the followingtheorem, which is a slightly weakened version of Theorem 12.

Theorem 56. Let (Xn, pn) be a sequence of proper length spaces convergingin the pointed Gromov–Hausdorff sense to a proper length space (X, p). As-sume there is a sequence of groups Γn acting discretely and almost transitivelyon Xn. If X is homeomorphic to a topological manifold, then X = N/Λ,where N is a simply connected nilpotent Lie group equipped with an invari-ant Finsler or sub-Finsler metric and Λ ≤ N is a central discrete subgroup.For large enough n, there are subgroups Λn ≤ π1(Xn, pn) and surjectivemorphisms

Λn → Λ = π1(X).

First, by Lemma 33, the isometry group of X acts transitively. So thefollowing theorem by Berestovskii tells us that its metric is either Finsler orsub-Finsler.

Theorem 57. ([5], Theorem 3). Let Y be a proper length space whoseisometry group acts transitively. If Y is homeomorphic to a topologicalmanifold, then its metric is given by a Finsler or a sub-Finsler structure.

Knowing this, we can start extracting information from the groups Γn.

Lemma 58. Let Y be a manifold equipped with a Finsler or sub-Finslermetric. Assume the group Γ acts on Y transitively by isometries. Thenthere is a constant K0 > 0 such that for every q ∈ Y , r ∈ (0, 1], there areelements γ1, . . . , γm ∈ Γ, m ≤ K0 such that

B(q, 3r) ⊂m⋃

j=1

B(γjq, r).

Proof. By ([23], Section 3), the Hausdorff dimension d of Y is an integer,and the corresponding Hausdorff measure µ on Y is positive on open setsand finite on compact sets. Furthermore, there is a constant N0 > 0 suchthat for all s ∈ (0, 1],

µ(B(q, 10s))

µ(B(q, s))≤ N0. (3)

Pick a maximal set γ1, . . . , γm ∈ Γ such that

20

γ1q, . . . , γmq ∈ B(q, 4r)

and the family of balls

B(

γjq,r2

)m

j=1

is disjoint. By Equation 3, m ≤ N0, and if the balls B(γjq, r)mj=1 do not

cover B(q, 3r), there is q′ ∈ B(q, 3r) with

minj=1,...,m

d(q′, γjq) > r.

Choosing γm+1 sending q to q′ contradicts the maximality of γ1, . . . , γm.

Now our goal is to apply a Generalized Margulis Lemma by Breuillard,Green and Tao to Γn to find large virtually nilpotent subgroups. Fix δ < 1and let

An(δ) := g ∈ Γn|d(gpn, pn) < (0.9)δ.

If we have good enough GH approximations between BXn(pn, 10) andBX(p, 10),

the balls BXn(pn, (2.1)δ) can be covered by K0 translates of BXn(pn, (0.9)δ),

where K0 comes from applying Lemmma 58 to X . This implies that (An(δ))2

can be covered by K0 translates of An(δ).

Definition 59. Let A be a finite symmetric subset of a group and K > 0.We say that A is a K-approximate group if there is a symmetric set W ⊂ A3

such that |W | ≤ K, A2 ⊂WA.

Fix M ∈ N. Then by Lemma 42, for large enough n there are symmetricgenerating sets Sn ⊂ Γn with SMn ⊂ An(δ).

Theorem 60. (Margulis Lemma, [7]) For each K > 1, there existsM ∈ N

such that the following holds. Let A be a K-approximate group in a groupG generated by a finite symmetric set S with SM ⊂ A. Then there existsubgroups H ⊳ G0 ≤ G such that:

• [G : G0] = OK(1).

• G0/H is nilpotent of nilpotency step OK(1).

• A4 ∩G0 generates G0 and contains H .

21

Remark 61. Theorem 60 represents one of the most important breakthroughsin modern mathematics. It shows that one can, after quotienting by a finitegroup, control the index and step of nilpotency in Gromov’s theorem ongroups of polynomial growth ([15]). Its proof idea is, on the surface, thesame as Gromov’s original proof. However, the techniques are much morerefined so that it took 30 years to achieve this improvement.

Applying Theorem 60 withK = K0 to An(δ), we get that for large enoughn, there are subgroups Hn(δ) ⊳ Gn(δ) ≤ Γn such that

• [Γn : Gn(δ)] ≤ I for some I ∈ N independent of n and δ.

• Gn(δ)/Hn(δ) is nilpotent of nilpotency step ≤ s for some s ∈ N inde-pendent of n and δ.

• Hn(δ) ⊂ g ∈ Γn|d(gpn, pn) < 10δ.

For each n, let δn ≥ 0 be the minimum number such that there are subgroupsH ′n ⊳ Γ

′n ≤ Γn such that

• [Γn : Γ′n] ≤ I.

• Γ′n/H

′n is nilpotent of nilpotency step ≤ s.

• H ′n ⊂ g ∈ Γn|d(gpn, pn) ≤ δn.

By the previous analysis, δn → 0 as n → ∞. Since [Γn : Γ′n] ≤ I, the

diameters diam(Xn/Γ′n) go to 0 as n→ ∞, so the groups Γ′

n also act almosttransitively.

Remark 62. The conclussion of Theorem 12 does not involve Γn, and thegroups Γ′

n also act almost trnasitively on the sequence Xn. This implies that

we can replace the groups Γn by the groups Γ′n and assume that every g ∈ Γ

(s)n

satisfies d(gpn, pn) ≤ δn.

4 Almost Translational Behaviour.

Let Γα be the equivariant ultralimit of the sequence Γn. In this section weuse techniques similar to the discrete ones in ([8], Chapter 3), [16] to identifyX with Γα and show that for large n, the action of Γn is almost translationalnear pn ∈ Xn.

22

Lemma 63. Let Γ be a group that acts ε-transitively by isometries on apointed metric space (Y, q0). Let H ⊳ G be a normal subgroup such that forevery h ∈ H , d(hq0, q0) ≤ ε0. Then

H ⊂ g ∈ G|d(gq, q) ≤ 2ε+ ε0 for all q ∈ Y .

Proof. Let h ∈ H , q ∈ Y . Then by ε-transitivity, there is g ∈ G such thatd(gq, q0) ≤ ε. Since H is normal, ghg−1 ∈ H , so

d(hq, q) = d(hg−1gq, g−1gq)

= d(ghg−1(gq), gq)

≤ d(ghg−1(gq), ghg−1(q0)) + d(ghg−1(q0), q0) + d(q0, gq)

≤ 2ε+ ε0.

Lemma 64. Γα is a nilpotent Lie group.

Proof. Since X is a locally compact homogeneous metric space of finite di-mension, by Theorem 14, Iso(X) is a Lie group. Since Γα is a closed subgroupof Iso(X), it is also a Lie group.

Since Γn acts εn-transitively onXn, Γα acts transitively onX . By Remark62, every g ∈ Γ

(s)α leaves p invariant. Then by Lemma 63, Γ

(s)α is trivial.

To show that Γα acts freely on X , we will require the following result byEnrico LeDonne and Alessandro Ottazzi.

Theorem 65. ([21], Theorem 1.2). Let G be a nilpotent Lie group equippedwith a left invariant Finsler or sub-Finsler metric, and f : G→ G an isometry.Then f is smooth, and uniquely defined by its first order data at the identity(f(e), df(e)).

Theorem 66. Γα is connected and acts freely on X .

Proof. Let Γ0 be the connected component of the identity in Γα. Since X isconnected, Γ0 acts transitively on X . For any point q ∈ X , its stabilizer

Γq := g ∈ Γ0|gq = q

is a compact subgroup. Since compact subgroups of connected nilpotent Liegroups are central, and all stabilizers are conjugate, all stabilizers coincide.

23

On the other hand, the action of Γ0 is faithful, so the stabilizers are trivial.By Theorem 57, after identifying X with Γ0, we can equip X with a nilpotentLie group structure such that its metric is invariant Finsler or sub-Finsler.By Theorem 65, the compact subgroup

Γ′p := g ∈ Γα|gp = p

consists of diffeomorphisms, hence there is an inner product 〈, 〉0 in TpXinvariant under Γ′

p. Then for each g ∈ Γ′p there is an 〈, 〉0-orthonormal basis

a1, b1, . . . , ak1, bk1 , c1, . . . , ck2, d1, . . . , dk3 ∈ TpX

and anglesθ1, . . . , θk1 ∈ S

1\1

such that

dpg(aj) = cos θjaj + sin θjbj ,

dpg(bj) = − sin θjaj + cos θjbj ,

dpg(cj) = −cj ,

dpg(dj) = dj.

Assume by contradiction that there exists g ∈ Γ′p\id, then again by Theo-

rem 65, dpg 6= id, so k1+k2 > 0. Let d0 denote the left invariant Riemannianmetric in X given by 〈, 〉0 and consider dU the uniform distance with respectto d0. Assume k1 > 0, then by the Baker–Cambell–Hausdorff formula, forevery ε > 0 there is δ > 0 such that

dU(Lexp(δa1), exp (+δa1) exp−1, Bd0(p, 100(2k1 + k2 + k3)δ)) < εδ

anddU(g, exp dpg exp

−1, Bd0(p, 100(2k1 + k2 + k3)δ)) < εδ.

Since s ≤ 2k1 + k2 + k3, by Lemma 23, there is C > 0 such that the step scommutators satisfy

dU([. . . [Lexp(δa1), g], . . .], g], exp [. . . [+δa1, dpg], . . .], dpg]exp−1, Bd0(p, δ)) < Cεδ.

However, by direct computation,

d0(exp [. . . [+δa1, dpg], . . .], dpg] exp−1(p), p) = δ|θ1 − 1|s + o(δ)

24

So, as δ → 0,[. . . [Lexp(δa1), g], . . .], g](p) 6= p,

contradicting the fact that every step s commutator is trivial. The casek2 > 0 is similar, but using c1 instead of a1.

Note that we have proved the following result.

Lemma 67. Let Y be a proper length space, and Γ ≤ Iso(Y ) be a closednilpotent subgroup acting transitively which is also a Lie group. Then Γ isconnected and acts freely.

By the above Lemmas, the identification Γα → X given by g → gp is ahomeomorphism. Therefore, X = Γα = N/Λ, where N is the universal coverof Γα equipped with an invariant Finsler or sub-Finsler metric and Λ ≤ N isa discrete central subgroup.

We have sequences εn → 0, Rn → ∞, and maps

fn : Xn → X, hn : X → Xn

such that fn(pn) = p, fn is an εn-approximation between BXn(pn, Rn) andBX(p, Rn), and

dU(fn hn, id, B(p, Rn)) < εn.

Definition 68. Let R > 1010 to be chosen later (Section 8) and define

Θn := g ∈ Γn|d(gpn, pn) < R,Θ′n := g ∈ Γn|d(gpn, pn) < R/2.

Define the translation maps t : Θn → X and t : Θn → Iso(X) as

t(g) := fn(g(hn(p))), t(g) := Lt(g).

The following two lemmas show that the maps t are approximate mor-phisms from Θn to Iso(X).

Lemma 69. For any R′ > 0 and any sequence gn ∈ Θn,

limn→α

dU(fn gn hn, t(gn), B(p, R′)) = 0.

25

Proof. By Lemma 66, the ultralimit gα of gn coincides with

L limn→α

t(gn) = limn→α

Lt(gn).

Therefore, by Lemma 31,

limn→α

dU(fn gn hn, limn→α

Lt(gn), B(p, R′)) = 0,

but also by Lemma 31,

limn→α

dU(Lt(gn), limn→α

Lt(gn), B(p, R′)) = 0.

The result follows by the triangle inequality of the uniform distance (Propo-sition 23).

Lemma 70. For any R′ > 0 and any pair of sequences gn, g′n ∈ Θ′

n, we have

limn→α

dU(t(gng′n), t(gn)t(g

′n), B(p, R′)) = 0.

Proof. By Proposition 23,

dU(t(gng′n), t(gn)t(g

′n), B(p, R′))

is bounded above by

dU(t(gng′n), fn gn g

′n hn, B(p, R′))

+ dU((fn gn) (g′n hn), (fn gn) (hn fn) (g

′n hn), B(p, R′))

+ dU((fn gn hn) (fn g′n hn), t(gn)t(g

′n), B(p, R′)).

Also by Proposition 23, the second summand goes to 0 as n → α, and thethird summand is bounded above by

dU(fn g′n hn, t(g

′n), B(p, R′))

+ dU(fn gn hn, t(gn), B(p, R′ + 2R))

By Lemma 69, all those summands go to 0 as n→ α.

Corollary 71. For any R′ > 0 and any pair of sequences gn, g′n ∈ Θ′

n, wehave,

limn→α

t(gng′n) = lim

n→αt(gn)t(g

′n) = lim

n→αt(gn) lim

n→αt(g′n).

26

5 Getting Rid of the Torsion.

Now we want to identify and get rid of the torsion elements of Γn. Unfortu-nately, for g ∈ Θn, t(g) being close to p, may not imply that high powers ofg do not “escape” Θn. That is because the definition of t : Θn → X containsan “error” coming from the fact that fn is not an actual isometry. To dealwith the torsion elements, we will need the escape norm from [7].

Since X is locally simply connected, there is ε0 ∈ (0, 1) such that everyloop of length ≤ 106ε0 is nullhomotopic. Let B be a small open convexsymmetric set in the Lie algebra n of Γα such that

exp(B) ⊂ B(p, ε0).

Define sets An and Sn as

An := g ∈ Θn|t(g) ∈ exp(B),

Sn := g ∈ Θn|t(g) ∈ exp(B/105K30 ),

An := An ∪ A−1n ,

Sn := Sn ∪ S−1n ,

where K0 ∈ N was obtained from Lemma 58.

Definition 72. [7]. Let A be a K-approximate group of a group G. We saythat A is a strong K-approximate group if there is a symmetric set S ⊂ Asatisfying the following:

• (asa−1|a ∈ A4, s ∈ S)103K3

⊂ A.

• If g, g2, . . . , g1000 ∈ A100, then g ∈ A.

• If g, g2, . . . , g106K3

∈ A, then g ∈ S.

By the Baker–Campbell–Hausdorff formula, Lemma 69, and Lemma 70,we see that if B was chosen small enough, for n sufficiently close to α,

asa−1|a ∈ A4n, s ∈ Sn ⊂ S2

n,

and all three conditions of a strong approximate group hold.

Lemma 73. For n sufficiently close to α, the set An, thanks to Sn, is astrong K0-approximate group of Γn.

27

Definition 74. Let A be a subset of a group G. For g ∈ G, we define theescape norm as

‖g‖A := inf

1

m+ 1| id, g, g2, . . . , gm ∈ A

.

In strong approximate groups, the escape norm satisfies really nice prop-erties.

Theorem 75. Let A be a strongK-approximate group. Then for g1, g2, . . . , gn ∈A10, we have

• ‖g1‖A = ‖g−11 ‖A.

• ‖gk1‖A ≤ |k|‖g1‖A

• ‖g2g1g−12 ‖A ≤ 103‖g1‖A.

• ‖g1g2 · · · gn‖A ≤ OK(1)∑n

i=1 ‖gi‖A.

• ‖[g1, g2]‖A ≤ OK(1)‖g1‖A‖g2‖A.

Proof. The first three properties are immediate. We refer the reader to ([7],Section 8) for a proof of the other two properties.

Lemma 73 and Theorem 75 imply that for n sufficiently close to α,

Hn := g ∈ An|‖g‖An = 0

is a subgroup of Γn normalized by An. By Lemma 42, for n sufficiently closeto α, Hn is normal in Γn and we can form the quotient Zn := Γn/Hn.

Remark 76. For any sequence hn ∈ Hn,

limn→α

t(hn) = p.

Thereforelimn→α

dGH(Xn/Hn, Xn) = 0.

What is special about the groups Zn is that they don’t have small sub-groups. Let Yn := π(An), where π : Γn → Zn is the standard quotientmap. For [g] ∈ Yn\eZn, we have ‖g‖An 6= 0, and therefore gM is not inA2n ⊃ AnHn for some M > 0. This implies that π(gM) = [g]M does not

belong to Yn. In other words, every nonidentity element in Yn eventually“escapes” from Yn.We still have the map

28

t : π(Θn) → X = Γα

given by

t([g]) := t(g).

Of course, to make this map well defined, we have to choose one representa-tive from each class in π(Θn). However, different choices of representativesonly change the value of t by an error which goes to 0 as n→ α.

Remark 77. For any pair of sequences gn, g′n ∈ Θn with π(gn) = π(g′n) for

α-large enough n, there is a sequence wn ∈ Hn such that gn = g′nwn forα-large enough n, and by Corollary 71,

limn→α

t(gn) = limn→α

t(g′n)t(wn) = limn→α

t(g′n).

Remark 78. By our choice of B, for α-large enough n, and g ∈ Y 8n ,

t(g) ∈ (exp(B))10 ⊂ BX(p, 10ε0) ∼= BN (id, 10ε0),

and we can think think of t as a map from Y 8n to N . We will denote this

map by tN : Y 8n → N .

6 The Nilprogressions.

We refer to ([7], Appendix B) for an introduction to local groups and mul-tiplicative sets. Local K-approximate groups are defined identically as K-approximate groups (Definition 59), but we replace the word group by mul-tiplicative set.

Definition 79. Let Am be a sequence of subsets of multiplicative sets Gm.If there is a K > 0 such that Am are local K-approximate groups for msufficiently close to α, we say that the algebraic ultraproduct A = limm→αAmis an ultra approximate group. If for α-large enough m, the approximategroups Am do not contain nontrivial subgroups, we say that A is a NSS (nosmall subgroups) ultra aproximate group.

For subsets A′m ⊂ A4

m with the property (A′m)

4 ⊂ A4m, we say that the

algebraic ultraproduct A′ =∏

m→αA′m is a sub-ultra approximate group if it

is an ultra approximate group, and there is a constant C0 ∈ N such that Amcan be covered by C0 many translates of A′

m for m sufficiently close to α.

29

Let Y be the algebraic ultraproduct∏

n→α Yn. Consider the map

t : Y 8 → X

given by the metric ultralimit

t(gn) := limn→α

t(gn).

Corollary 71 implies that t is a homomorphism, but moreover, it is a good

model.

Definition 80. [18], [7]. Let A =∏

m→αAm be an ultra approximate group.A good Lie model for A is a connected local Lie group L, together with amorphism σ : A8 → L satisfying:

• There is an open neighborhood U0 ⊂ L of the identity with U0 ⊂ σ(A)and σ−1(U0) ⊂ A.

• σ(A) is precompact.

• For F ⊂ U ⊂ U0 with F compact and U open, there is an algebraicultraproduct A′ =

∏

m→αA′m of finite sets A′

m ⊂ Am with σ−1(F ) ⊂A′ ⊂ σ−1(U).

Definition 81. LetB be a local group, u1, u2, . . . , ur ∈ B, andN1, N2, . . . , Nr

∈ R+. The set P (u1, . . . , ur;N1, . . . , Nr) is defined as the set of words in theui’s and their inverses such that the number of appearances of ui and u−1

i

is not more than Ni. We say that P (u1, . . . , ur;N1, . . . , Nr) is well defined ifevery word in it is well defined in B. We call it a progression of rank r. Wesay it is a nilprogression in C-regular form for some C > 0 if it also satisfiesthe following properties:

• For all 1 ≤ i ≤ j ≤ r, and all choices of signs, we have

[u±1i , u±1

j ] ∈ P

(

uj+1, . . . , ur;CNj+1

NiNj

, . . . ,CNr

NiNj

)

.

• The expressions un11 . . . unr

r represent distinct elements as n1, . . . , nrrange over the integers with |n1| ≤ N1/C, . . . , |nr| ≤ Nr/C.

30

For a nilprogression P in C-regular form, and ε ∈ (0, 1), it is easy to seethat P (u1, . . . , ur; εN1, . . . , εNr) is also a nilprogression in C-regular form.We denote it by εP . We define the thickness of P as the minimum ofN1, . . . , Nr and we denote it by thick(P ). The set un1

1 . . . unrr ||ni| ≤ Ni/C

is called the grid part of P , and is denoted by G(P ).Let Pm be a sequence of sets. If for α-large enough m, Pm is a nilprogres-

sion of rank r in C-regular form for some r ∈ N, C > 0, independent ofm, wesay that the algebraic ultraproduct P =

∏

m→α Pm is an ultra nilprogression

of rank r in C-regular form. We denote∏

m→α εPm as εP . If (thick(Pm))mis unbounded, we say that P is a nondegenerate ultra nilprogression. Thealgebraic ultraproduct G(P ) :=

∏

m→αG(Pm) is called the grid part of P .

The goal of this section is to obtain the following theorem.

Theorem 82. Let A =∏

m→αAm be a local NSS ultra approximate group.Assume there is a good Lie model σ : A8 → L. Then A4 contains a nonde-generate ultra nilprogression P of rank r := dim(L) in C-regular form, withthe property that for all standard ε ∈ (0, 1), there is an open set Uε ⊂ Lwith σ−1(Uε) ⊂ G(εP ).

Proof. In the proof of ([7], Theorem 9.3), they construct via the short basistrick of [8], [16] and an induction on dim(L), a set P and they prove it isa nondegenerate ultra nilprogression of rank dim(L) in C-regular form. Wewill repeat that construction and show that for every standard ε ∈ (0, 1),there is an open set Uε ⊂ L with σ−1(Uε) ⊂ G(εP ).

Let B be a small convex set in l, the Lie algebra of L. Let A′, A′′, A′′′ besub ultra approximate groups of A such that

σ−1(exp(B)) ⊂ A′ ⊂ σ−1(exp((1.001)B)),σ−1(exp(δB)) ⊂ A′′ ⊂ σ−1(exp((1.001)δB)),

σ−1(exp(δB/10)) ⊂ A′′′ ⊂ σ−1(exp((1.001)δB/10)),

where δ > 0 will be chosen later. Let u ∈ A′\id be such that minimizes‖u‖A′ (in this setting, ‖·‖A′ is a nonstandard real number). Let Z = un||n| ≤1/‖u‖A′. The image σ(Z) is of the form φ([−1, 1]), for some φ : [−1, 1] → L,φ(t) = exp(tv), v ∈ l (see [7], Theorem 9.3).

If B is small enough, then for all x ∈ (A′′)10, ‖x‖A′′ ≤ 100δ, and then, byTheorem 75, if δ was chosen small enough,

‖[u, x]‖A′ = O(‖u‖A′‖x‖A′) < ‖u‖A′ for all x ∈ (A′′)10.

31

Since ‖u‖A′ was minimal, u commutes with every element in (A′′)10. Thatimplies that Z commutes with everyone in (A′′)10 and we can form the quo-tients

π : A′′ → A′′/Z and π : A′′′ → A′′′/Z.

Also, σ(Z) commutes with everyone in exp(δB), so by the connectednessof L, σ(Z) lies in the center of L. For some open

exp(9δB) ⊂ U ⊂ exp(10δB),

we can form the local quotient π : U → U/σ(Z). Then

πσ : (A′′′/Z)8 → U/σ(Z)

is a good Lie model and A′′/Z has the NSS property (see [7], Theorem9.3), so we can apply the induction hypothesis and conclude that there isa nondegenerate P (u1, . . . , ur−1;N1, . . . , N r−1) in C-regular form such thatP ⊂ (A′′′/Z)4 ⊂ A′′/Z and for every ε > 0, there is an open Wε ⊂ U/σ(Z)with (πσ)−1(Wε) ⊂ G(εP ). To properly “lift” P to A′′, the following lemmais required.

Lemma 83. (Lifting Lemma). For every w ∈ A′′/Z, there is w′ ∈ A′′ withπ(w′) = w, and ‖w′‖A′′ = O(‖w‖A′′/Z).

The proof of the lemma can be found in ([7], Theorem 9.5).Construct P (u1, . . . , ur; N1, . . . , Nr), where ui ∈ A′′ is a lift of ui that

minimizes ‖ui‖A′′ for i = 1, . . . , r − 1,

Ni := δ0N i for i = 1, . . . , r − 1,ur := u, Nr := δ0/‖u‖A′′.

For some small enough standard δ0 > 0, P is a nondegenerate ultranilpro-gression in regular form (see [7], Theorem 9.3). We need to check that for allε > 0, there is an open Uε ⊂ L such that σ−1(Uε) ⊂ G(εP ).

By contradiction, assume that for some ε > 0, the element x of A′′\G(εP )with minimal norm ‖x‖A′′ satisfies σ(x) = idL. If that is the case, π(σ(x)) =idL/σ(Z), and by our induction hypothesis, for all standard η > 0, πx ∈G(ηP ). Therefore x = un1

1 . . . unrr , with

|ni| ≤ ηNi/C for i = 1, . . . , r − 1, |nr| ≤ 1/‖ur‖A′ .

32

Also, using Theorem 75, and the fact that N i = O(1/‖ui‖A′′/Z), we get

‖un11 . . . u

nr−1

r−1 ‖A′′ = O

(

r−1∑

i=1

‖uni

i ‖A′′

)

= O

(

r−1∑

i=1

|ni|‖ui‖A′′

)

= O

(

ηr−1∑

i=1

Ni‖ui‖A′′/Z

)

= O(η).

Since η was arbitrary, we obtain that ‖un11 . . . u

nr−1

r−1 ‖A′′ is infinitesimal.Using once more Theorem 75,

‖unrr ‖A′′ = O(‖x‖A′′ + ‖un1

1 . . . unr−1

r−1 ‖A′′).

This implies that ‖unrr ‖A′′ is infinitesimal and |nr| = o(Nr) ≤ εNr/C.

Also, since η was arbitrary, |ni| ≤ εNi/C for i = 1, . . . , r − 1. Thereforex ∈ G(εP ), which is a contradiction.

Remark 84. Note that from the proof of Theorem 82, the group L is nilpo-tent and the basis l1, . . . , lr of l given by

exp(li) = σ(

u⌊Ni/C⌋i

)

for i = 1, . . . , r,

is a Malcev basis (see [11] for the definition of a Malcev basis).

Let r := dim(X). Applying Theorem 82 to t : Y 8 → X , we obtain thefollowing.

Proposition 85. There is C > 0, and for each n ∈ N, elements u1, . . . , ur ∈Zn, N1(n), . . . , Nr(n) ∈ N with

limn→α

Ni(n) = ∞ for each i = 1, . . . , r,

satisfying that for every ε ∈ (0, 1), there is δ > 0 such that for α-large enoughn,

Pn := P (u1, . . . , ur;N1(n), . . . , Nr(n))

is a nilprogression in C-regular form, and

g ∈ π(Θn)|d(t(g), p) < δ ⊂ G(εPn) ⊂ g ∈ π(Θn)|d(t(g), p) < 10ε0.

33

7 Malcev Theory.

Let P (v1, . . . , vr;N1, . . . , Nr) be a nilprogression in C-regular form. As-sume Ni ≥ C for each i, and set ΓP to be the abstract group generated

by γ1, . . . , γr with relations [γi, γj] = γβj+1i,j

j+1 . . . γβri,jr whenever i < j, where

[vi, vj ] = vβj+1i,j

j+1 . . . vβri,jr and |βli,j| ≤ CNl

NiNj. We say that P is good if ΓP is

isomorphic to a lattice in a simply connected nilpotent Lie group, and eachelement of ΓP has a unique expression of the form

γn11 . . . γnr

r , with n1, . . . , nr ∈ Z.

Theorem 86. (Malcev Embedding) Let r ∈ N, C > 0, and P (v1, . . . , vr;N1, . . . , Nr) a nilprogression in C-regular form in a group Γ. If thick(P ) islarge enough depending on r, C, then P is good and the map vi → γi extendsto an embedding ♯ : G(P ) → ΓP . For A ⊂ G(P ), we will denote its imageunder this embedding by A♯. Furthermore, there is a quasilinear polynomialgroup structure (see Definition 40)

Q : Rr × Rr → R

r

of degree ≤ d(r) such that the multiplication in ΓP is given by

γn11 . . . γnr

r γm11 . . . γmr

r = γ(Q(n,m))11 . . . γ

(Q(n,m))rr for n,m ∈ Zr.

Q is called the Malcev polynomial of P , and (Rr, Q) the Malcev Lie group ofP . ΓP is isomorphic, via γi → ei, to the lattice (Zr, Q|Zr×Zr).

The proof can be found in ([10], Section 4.2).By Theorem 86, for α-large enough n, the nilprogressions Pn are good

with Malcev polynomials Qn. Defne the maps

t: G(Pn)

♯ → X and tN : G(Pn)

♯ → N

as

t(x♯) = t(x) and t

N(x

♯) = tN(x).

Lemma 87. Let r ∈ N, C > 1, then there exist M0, δ0 > 0 such that, for anynilprogression P of rank r in C-regular form with thick(P ) > M0, we have

G(δ0P )2 ⊂ G(P ).

34

The proof can be found in ([7], Appendix C).

Definition 88. Let δ0 > 0 be given by Lemma 87 with r, C from Proposition85. For n, the Lie algebra of N , let v1, . . . , vr be the basis such that

exp(vi) := limn→α

t

(

u

⌊

δ0Ni(n)

C

⌋

i

)

for each i = 1, . . . , r.

By Remark 84, v1, . . . , vr is a Malcev basis, and the map ψ : Rr → N givenby

ψ(x1, . . . , xr) := exp(x1v1) . . . exp(xrvr)

is a diffeomorphism.

Lemma 89. The map Rr × Rr → Rr given by

(x, y) → ψ−1(ψ(x)ψ(y))

is a quasilinear polynomial of degree ≤ d(r).

Proof. By the Baker–Campbell–Hausdorff formula, after identifying n withRr via the basis v1, . . . , vr, the map Rr × Rr → Rr given by

(x, y) → exp−1(ψ(x)ψ(y))

is polynomial of degree ≤ r. Also, the map Rr → Rr given by

x→ ψ−1(exp(x))

is polynomial of degree bounded by a number depending only on r (see [11],Section 1.2). Therefore the composition is also polynomial of degree ≤ d(r).Quasilinearity is immediate from the definition.

Let N0 ∈ N be given by Lemma 39 with d(r) given by Lemmas 86 and89, and δ0 > 0 as in Definition 88. Set ξ : N → N as

ξ(n) := N0

⌊

δ0n

CN0

⌋

.

For n ∈ N, consider κn : Rr → Rr given by

κn(x1, . . . , xr) := (x1ξ(N1(n)), . . . , xrξ(Nr(n))).

35

Let Gn be the group (Rr, Qn), where Qn : Rr×Rr → Rr is the group structuregiven by

Qn(x, y) := κ−1n (Qn(κn(x), κn(y))).

Note that after identifying ΓPn with Zn, the map κ−1n : ΓPn → Gn is an injec-

tive morphism. For A ⊂ ΓPn, we will denote its image under this embeddingby A. Also define

Ω :=

−1, . . . ,−1

N0

, 0,1

N0

, . . . , 1

×r

⊂ Rr.

Consider the maps ωn : Ω → ΓPn defined as

ωn(x1, . . . , xr) = γx1ξ(N1(n))1 . . . γ

xrξ(Nr(n))r .

Finally define ωα : Ω → N as ωα = ψ|Ω. Consider the following diagram.

Ω× Ω (G(δ0Pn)♯)×2 G(Pn)

♯ Rr

Ω× Ω N ×N N Rr

ωn

id

∗

tN t

N

κ−1n

id

ωα ∗ ψ−1

The first row of the diagram is the polynomial Qn, while the second rowis the polynomial Q. Commutativity of the diagram does not hold in general,but it holds in the limit, as the following proposition shows.

Proposition 90. For every x, y ∈ Ω,

limn→α

κ−1n (ωn(x)ωn(y)) = ψ−1(ωα(x)ωα(y)). (4)

Proof. We will first show that for any sequence x♯n ∈ G(Pn)♯, we have

limn→α

κ−1n (x♯n) = lim

n→αψ−1(t

N (x

♯n)). (5)

We can decompose the sequence xn = upn,1

1 . . . upn,rr ∈ G(Pn) as

xn = xn,1 . . . xn,r, with xn,j = upn,j

j .

And by Lemma 71,

limn→α

tN(xn) = limn→α

tN(xn,1) . . . limn→α

tN(xn,r)

= exp

(

limn→α

Cpn,1δ0N1(n)

v1

)

. . . exp

(

limn→α

Cpn,rδ0Nr(n)

vr

)

.

36

On the other hand, by definition,

limn→α

κ−1n (xn) =

C

δ0limn→α

(

pn,1N1(n)

, . . . ,pn,rNr(n)

)

.

Henceψ(

limn→α

κ−1n (xn)

)

= limn→α

tN(xn),

establishing Equation 5. Finally, for x, y ∈ Ω,

limn→α

tN(ωn(x)ωn(y)) = limn→α

tN(ωn(x)) limn→α

tN (ωn(y))

= ωα(x)ωα(y).

Combining this with Equation 5, we obtain Equation 4.

Remark 91. By Proposition 90 and Lemma 39, we see that Qn convergeswell to Q, and by Lemma 38, the corresponding Lie algebras converge wellto n.

8 Discrete Subgroups.

In this section we finish the proof of Theorem 56 by establishing the followingproposition.

Proposition 92. For α-large enough n, there are groups Λn ≤ π1(Xn) andsurjective morphisms Λn → Λ.

Recall that Zn = Γn/Hn. Let η > 0 be such that for α-large enough n,

Dn := g ∈ Zn|d(g(pnHn), pnHn) < η ⊂ G(δ0Pn).

Let Zn be the abstract group generated by Dn, with relations

s = s1s2 ∈ Zn whenever s, s1, s2 ∈ Dn and s = s1s2 in Zn.

Remark 93. By Theorem 86, for α-large enough n, Zn = ΓPn, and byTheorem 44, there is a regular η/3-wide covering map X ′

n → Xn/Hn whoseGalois group is the kernel of the canonical map Φn : ΓPn → Zn.

37

Remark 94. By Theorem 86, Ker(Φn)∩D♯n = eΓPn

, and for every g ∈ ΓPn

withd(Φn(g)(pnHn), pnHn) < ε < η,

there is w ∈ G(Pn)♯ such that

d(Φn(w)(pnHn), pnHn) < ε,

and gw ∈ Ker(Φn).

Let λ1, . . . , λℓ → Λ ≤ N be a basis of Λ as a free abelian group. SinceN is a simply connected nilpotent Lie group, there is M ∈ N such that theM-th roots of the λi lie in B

N (p, η/2). For each i ∈ 1, . . . , ℓ pick a sequence

λi(n) ∈ G(Pn)♯ ⊂ ΓPn

≤ Gn = (Rr, Qn)

such that

limn→α

tN (λi(n)) =

λiM.

By Equation 5,

limn→α

λi(n) = ψ−1

(

λiM

)

.

Since Qn converges well to Q,

limn→α

(λi(n))M = ψ−1(λi).

Therefore, if R from Definition 68 was chosen large enough,

limn→α

t(

Φn(

λi(n)M))

= limn→α

t(

(Φn (λi(n)))M)

=(

limn→α

t (Φn (λi(n))))M

=(

limn→α

t(λi(n))

)M

=

(

λiM

)M

= eX .

By Remark 94, for α-large enough n, there are wn,i ∈ G(Pn)♯ such that

λi(n)Mwn,i ∈ Ker(Φn),

38

andlimn→α

t(Φn(wn,i)) = limn→α

t(wn,i) = eX .

Since Qn converges well to Q,

limn→α

λi(n)Mwn,i = λi for each i ∈ 1, . . . , ℓ,

by Remark 91 and Lemma 41,

limn→α

logn(λi(n)Mwn,i) = log(λi) for each i ∈ 1, . . . , ℓ,

where logn, log, denote the logarithm maps with respect to Qn and Q, re-spectively. Therefore for α-large enough n, the set

logn(λ1(n)Mwn,1), . . . , logn(λℓ(n)

Mwn,ℓ)

is linearly independent. Also, for i, j ∈ 1, . . . , ℓ,

limn→α

[λi(n)Mwn,i, λj(n)

Mwn,j] = [ limn→α

λi(n)Mwn,i, lim

n→αλj(n)

Mwn,j]

= λ−1i λ−1

j λiλj

= eN .

By Remark 94, for α-large enough n,

[λi(n)Mwn,i, λj(n)

Mwn,j] = eΓPn,

and the group

〈λ1(n)Mwn,1, . . . , λℓ(n)

Mwn,ℓ〉 ≤ Ker(Φn)

is a free abelian group of rank ℓ. This implies that there is a subgroupΛn ≤ Ker(Φn) isomorphic to Λ. By Theorem 46, Remark 76, and Remark93, for α-large enough n, there is a regular covering Xn → Xn with Galoisgroup Ker(Φn), so there is a surjective map

π1(Xn, pn) → Ker(Φn).

Letting Λn ≤ π1(Xn, pn) be the preimage of Λn, we have surjective maps

Λn → Λ.

39

9 Nilpotent Locally Compact Groups of Isome-

tries.

The goal of this section is to prove the following theorem, which combinedwith Theorem 56, yields Theorem 12.

Theorem 95. Let (Xn, pn) be a sequence of proper length spaces converg-ing in the pointed Gromov–Hausdorff sense to a proper length space (X, p).Assume there is a sequence of groups Γn acting discretely and almost transi-tively on Xn. If X is semilocally simply connected, then it is homeomorphicto a topological manifold.

We start by proving a Lemma analogous to Lemma 58 in the infinitedimensional setting.

Lemma 96. Let (Y, q) be a proper pointed length space, and Γ a groupacting transitively by isometries. Then for every r > 0, there are elementsγ1, . . . , γK0 ∈ Γ, with K0 depending on r, such that

B(q, 3r) ⊂

K0⋃

j=1

B(γjq, r).

Proof. The collection of balls B(γq, r)γ∈Γ is such that the union of theirinteriors covers B(q, 3r). By compactness of B(q, 3r), finitely many suffice.

Fix 0 < δ < 1 and let

An(δ) := g ∈ Γn|d(gpn, pn) < (0.9)δ.

If we have good enough GH approximations between BXn(pn, 10) andBX(p, 10),

the ballsBXn(pn, (2.1)δ) can be covered byK0(δ) translates ofBXn(pn, (0.9)δ),

where K0(δ) comes from applying Lemmma 58 to X . This implies that(An(δ))

2 can be covered by K0 translates of An(δ).For any fixed M ∈ N, by Lemma 42, if n is large enough, there are

symmetric generating sets Sn ⊂ Γn with SMn ⊂ An(δ). Applying Theorem 60with K = K0 to An(δ), we get that for large enough n, there are subgroupsGn(δ) ≤ Γn such that

• [Γn : Gn(δ)] ≤ I for some I(δ) ∈ N independent of n.

40

• Gn(δ)(s) ⊂ g ∈ Gn(δ)|d(gpn, pn) < 10δ for some s(δ) ∈ N indepen-

dent of n.

Since [Γn : Gn(δ)] ≤ I, the squence of groups Gn(δ) also acts almosttransitively. Since the conclussion of Theorem 95 does not involve the groupsΓn, we can replace them by the groups Gn(δ) and assume that

Γ(s)n ⊂ g ∈ Γn|d(gpn, pn) < 10δ

for all large enough n and some s depending on δ.

Remark 97. Iterating this process with distinct δ, for proving Theorem 95,we can assume that for every k ∈ N, there is s(k) ∈ N such that for all largeenough n, we have

Γ(s)n ⊂ g ∈ Γn|d(gpn, pn) ≤ 1/2k+1.

Which, by Lemma 63, implies for large enough n,

Γ(s)n ⊂ g ∈ Γn|d(gq, q) ≤ 1/2k for all q ∈ Xn.

Let Γα be the equivariant ultralimit of the sequence Γn. We identifyit with a closed subgroup of Iso(X) acting transitively. We will now useAndrew Gleason and Hidehiko Yamabe solution of Hilbert’s fifth problem tofind nilpotent Lie groups close to X .

Theorem 98. [13], [29]. Let Γ be a locally compact Hausdorff group. Thenthere is an open subgroup G such that for any neighborhood U of the identity,there is a compact normal subgroupH⊳G withH ⊂ U , andG/H a connectedLie group.

Let G ≤ Γα be the open subgroup given by Theorem 98. By the followingtheorem of Berestovskii, G still acts transitively on X .

Theorem 99. ([3], Theorem 1). Let Y be a proper length space, and Γ aclosed subgroup of Iso(Y ) acting transitively. If G ≤ Γ is an open subgroup,then G acts transitively as well.

Since open subgroups are also closed, G is still a locally compact group.For every k ∈ N, set

Uk := g ∈ G|d(gp, p) ≤ 1/2k.

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Define H1 ⊳ G to be a compact subgroup with H1 ⊂ U1, and G/H1 a Liegroup. We would like to define inductively a sequence of nested compactsubgroups

. . . ⊳ Hk+1 ⊳ Hk ⊳ . . . H1 ⊳ G

such that Hk is normal in G, Hk ⊂ Uk, and G/Hk is a Lie group for everyk. We can do it thanks to our choice of G and the following theorem byViktor Glushkov.

Theorem 100. [14]. Let G be a locally compact group, and H1, H2 be twoclosed normal subgroups such that G/H1 and G/H2 are both Lie groups.Then G/(H1 ∩H2) is also a Lie group.

Let ρk : G → Iso(X/Hk) be the induced action. Note that Ker(ρk+1) ≤Ker(ρk) and Ker(ρk) ⊂ Uk for each k, so we can replace Hk by Ker(ρk) andassume that G/Hk acts faithfully on X/Hk for each k.

Let Gk := G/Hk and πk : G → Gk be the canonical projection. SinceGk is a Lie group, it has a neighborhood Wk of the identity that contains nonontrivial subgroups. If Wk := (πk)−1(Wk), then every subgroup containedin Wk is automatically contained in Hk. If ℓ ∈ N is large enough,

g ∈ G|d(gq, q) ≤ 1/2ℓ for all q ∈ X ⊂ Wk,

so by Remark 97, there is s ∈ N, depending on k, such that G(s) ⊂ Hk,and Gk is nilpotent. By Lemma 67, Gk acts freely on X/Hk.

Lemma 101. G acts freely on X .

Proof. If gq = q for some g ∈ G, q ∈ X , then for each k ∈ N, gHk ∈ Gk fixesthe class of q in X/Hk. Since Gk acts freely, g ∈ Hk. This holds for every k,so

g ∈∞⋂

k=1

Hk = e.

Remark 102. By Lemma 101, we can identify G with X .

Let s1 ∈ N be such that G(s1)1 = e. For every k ∈ N, there is a continu-

ous surjective morphism Gk → G1 with compact kernel H1/Hk. This implies

that G(s1)k ≤ H1/Hk, and since compact subgroups of connected nilpotent

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Lie groups are central, G(s1+1)k = e. Therefore, G(s1+1) ≤ Hk for all k ∈ N,

and G(s1+1) = e, so G is nilpotent and Theorem 11 is thereby proved.To finish the proof of Theorem 95, we need to consider two separate cases.

The first case is when for only finitely many k, the connected component ofHk/Hk+1 is nontrivial. In this case, let k0 be such that Hj/Hj+1 is discretefor all j ≥ k0. Let δ > 0 be small enough so that BGk0

(e, δ) contains nonontrivial subgroups and BGk0

(e, δ) → Gk0 has no content. By Lemma 49and induction, the balls BGk

(e, δ) contain no nontrivial subgroups if k ≥ k0.For ℓ ∈ N large enough,

Hℓ/Hℓ+1 ⊂ g ∈ Gℓ+1|d(g, e) ≤ 1/2ℓ+1 ⊂ BGℓ+1(e, δ),

hence the groups Hℓ are trivial. This implies that G = Gℓ for all suffi-ciently large ℓ, and G is a Lie group.

We are left to the case when for infinitely many k, the connected compo-nents of the groups Hk/Hk+1 are nontrivial. We will assume this, which willultimately contradict the semilocal simple connectedness of G.

Since G is semilocally simply connected, there is δ > 0 such that theinclusion BG(e, δ) → G has no content. By our assumption, there is k ∈ N

with 1/2k−100 ≤ δ, and the connected component of Hk−1/Hk is nontrivial.By Lemma 48, there is a nontrivial loop in Gk based at e whose image iscontained in BGk

(e, 1/2k−2). Our contradiction will arise from lifting thisloop to a nontrivial small loop in G.

By Lemma 52, there is a noncontractible loop γ0 : [0, 1] → Gk based at ewith length(γ0) ≤ 1/2k−4. Since Gk is semilocally simply connected, there isε > 0 such that any two closed curves in Gk whose uniform distance to eachother is less than ε are automatically homotopic to each other.

Choose ℓ ∈ N large enough so that 1/2ℓ−10 ≤ ε and let f : Gℓ → Gk

be the natural map, which by Lemma 54 is a metric submersion. UsingLemma 55, we get a curve γ1 : [0, 1] → Gℓ with γ1(0) = e, f γ1 = γ0, andlength(γ1) ≤ 1/2k−4. Since f(γ1(1)) = e, we have γ1(1) ∈ Hk/Hℓ. For eachm ∈ N, define the curve βm : [0, 1] → Gℓ as βm(t) := γ1(1)

m−1γ1(t). Observethat βm(1) = βm+1(0), so we can define the curves γm as βm ∗ . . . ∗ β1.

Let m0 be the smallest integer such that γm0(1) lies in the connectedcomponent of the identity of Hk/Hℓ, and let β : [0, 1] → Hk/Hℓ be a (notnecessarily rectifiable) curve from γm0(1) to e. Then the curve β ∗ γm0 is acurve whose image lies in BGℓ

(e, 1/2k−6), and the composition f (β ∗ γm0) ishomotopic to γ0∗ . . .∗γ0 (m0 times). Since π1(Gk) has no torsion, f (β∗ γm0)

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is noncontractible in Gk. By Lemma 51, there is a loop γ in Gℓ based at ewith length(γ) ≤ 1/2k−8, such that f γ is noncontractible in Gk.

By Lemma 55, there is a curve c1 : [0, 1] → BG(e, 1/2k−6) with c1(0) = e,

c1(1) ∈ Hℓ, and πk c1 noncontractible in Gk. Take a minimizing path c2from c1(1) to e ∈ G. Then the curve c := c2 ∗ c1 is a loop in G satisfying thatc([0, 1]) ⊂ BG(e, δ/2), and

dU(πk c, πk c1, [0, 1]) ≤ 1/2ℓ−2

for some suitable reparametrization c1 of c1. This implies that πk c is non-contractible in Gk, and the composition BG(e, δ) → G → Gk has nontrivialcontent, which is a contradiction.

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