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On the Geometry of Raysand the Gromov Compactificationof Negatively Curved Manifolds

Andrea Sambusetti - Sapienza Università di Roma, Italy

joint work with: F. Dal’bo - University of Rennes, FranceM. Peigné - University of Tours, France

Contributions in Differential Geometrya round table in occasion of the 65th birthday of L. Bérard Bergery

Université du Luxembourg - 6/9 September, 2010

History“Näives” Compactifications

The Gromov CompactificationThe Busemann map

Geometrically finite manifolds

Outline

1 History

2 “Näives” Compactifications

3 The Gromov Compactification

4 The Busemann map1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence

5 Geometrically finite manifoldsGeneral resultsExample in dimension n = 3

Rays, parallels, Busemann functions...

XIX c. Beltrami : Desargues spaces• the only simply connected are En, Sn,Hn

late XIX c. Hadamard : nonpositively curved surfaces in E3

• unicity of geodesics in a homotopy class• ends (cusps, funnels)• rays asymptotic to cusps and funnels

1920’s Cartan : generalization to higher dimension (Cartan-Hadamard manifolds)• expo : ToX → X diffeomorphism• no geodesic loops, no critical points...

1940’s Busemann : Desargues geodesic spaces• theory of parallels & Busemann functions

1970’s Gromov : functional compactification of general Riemannian manifolds

Some other applications of the Busemann functions:Soul Theorem (Cheeger-Gromoll-Meyer), Toponogov’ Splitting Theorem,Harmonic and (noncompact) Symmetric spaces, dynamics of Kleinian groups...

Compactifying by adding “directions”

Example. X = En

∂X = {half-lines}/parallelism ∼= Sn−1

X = X ∪ ∂X ∼= Bn−1

Example. X = En

X = X ∪ ∂X ∼= Bn−1

Näif idea: X general, complete Riemannian manifold

Example. X = En

X = X ∪ ∂X ∼= Bn−1

R(X ) = {rays of X} α : R+ → X is a ray if it is globally minimizingi.e. `(α; s, t) = d(α(s), α(t)) for all s, t ≥ 0

Example. X = En

X = X ∪ ∂X ∼= Bn−1

R(X ) = {rays of X}∂X = R(X )/“parallelism”

X = X ∪ ∂X(with some reasonable topology to be defined)

α : R+ → X is a ray if it is globally minimizingi.e. `(α; s, t) = d(α(s), α(t)) for all s, t ≥ 0

parallelism =?(on general Riemannian manifolds)

Compactifying by adding “directions”Parallelism for rays on a general, complete Riemannian manifold X :

α and β are metrically asymptotic (i.e. d∞(α, β) <∞)if supt≥0 d(α(t), β(t)) < +∞

α tends visually to β (i.e. α � β) –β coray to α–

if−−−−−−→β(0)α(tn)→ β for some {tn} → ∞

−→xy = a minimizing geodesic segment from x to y

if−−−−→bnα(tn)→ β for some {tn} → ∞, {bn} → β(0)

Technical fact: the correct definition of visual convergenceasks for a sequence bn → β(0) with

−−−−→bnα(tn)→ β

Technical fact: the correct definition of visual convergenceasks for a sequence bn → β(0) with

−−−−→bnα(tn)→ β

Otherwise, on a spherical-capped cylinder with pole o,different meridians would not be visually equivalent rays from o!

(not symmetric, apriori)

α and β are metrically asymptotic (d∞(α, β) <∞)if supt≥0 d(α(t), β(t)) < +∞

α and β are visually asymptotic from o (α ≺o� β )if ∃ a ray γ from o such that α � γ and β � γ

That is:α ≺o� β if one can see (asymptotically) α and βunder a same direction from o

(depends on o, apriori)

Theorem. [Folklore: Busemann, Shihoama et al.]

Two rays α and β are visually asymptotic from every point o| {z } iff Bα = Bβ

“visually asymptotic”

Two rays α and β are visually asymptotic from every point o| {z } iff Bα = Bβ

“visually asymptotic”

Bα = the Busemann function of the ray α

Compactifying by adding “directions”Busemann function of a ray on a general, complete Riemannian manifold X :

Bα(x , y) = limt→+∞

xα(t)− α(t)y

(asymptotic defect of triangles xypwith third vertex p = α(t), t � 0)

xα(t)− α(t)y

Fact: always converges

xα(t)− α(t)y

Example: X = En

−→xy parallel to α ⇔ ∠α(t)x , yt→∞−→ 0

(as x , y are fixed) ⇔ xy + yα(t)− xα(t)→ 0⇔ Bα(x , y) = d(x , y)

xα(t)− α(t)y

Folklore: X = any Riemannian manifold

Bα(x , y) = d(x , y) iff (−→xy is a ray and) α � −→xy .

xα(t)− α(t)y

⇓if Bα = Bβ then α � γ iff β � γ for every γ (and reciprocally)

xα(t)− α(t)y

⇓if Bα = Bβ then α � γ iff β � γ for every γ, which implies

Two rays α and β are visually asymptotic from every point o iff Bα = Bβ .

Compactifying by adding “directions”X = general, complete Riemannian manifold∂X .

= ∂mX = R(X )/(d∞(α,β)<+∞) i.e. modulo metric asymptoticityor

∂X .= ∂v X = R(X )/(Bα=Bβ ) i.e. modulo visual asymptoticity

X = X ∪ ∂X compactifies? what topology?

X = X ∪ ∂X compactifies? what topology? R(X) has the uniform topology (u.c. on compacts)which means: αn → α iff α′n(0)→ α′(0).

Theorem. [Folklore: Eberlein, O’ Neill...]

Let X be a Cartan-Hadamard manifold(i.e. complete, simply connected with k(X) ≤ 0).(i) d∞(α, β) < +∞ ⇔ Bα = Bβ

so X(∞).

= ∂mX = ∂v X(ii) X has a natural “visual” topology

(xn→ξ∈X(∞) iff ∠oxn,α(n)→ 0 ∃α∈ξ, ∃o∈X )

s.t. X ↪→ X is a topological embedding(iii) X is a compact, metrizable space

X(∞) is called the visual boundary.

R(X) has the uniform topology (u.c. on compacts)which means: αn → α iff α′n(0)→ α′(0).

so X(∞).

Dropping the “simply connected” assumption?

R(X) has the uniform topology (u.c. on compacts)which means: αn → α iff α′n(0)→ α′(0).

so X(∞).

Examples: flutes and ladders [DPS]

There exist hyperbolic manifolds Xwith rays α, β and αn → α

in each of the following situations:

(a) α � β and β � α but Bα 6= Bβ

(b) d∞(α, β) <∞ but Bα 6= Bβ

(c) d∞(α, β) =∞ but Bα = Bβ

(d) d∞(αn, αm) <∞ and Bαn = Bαm∀n,mbut d∞(αn, α) =∞ and Bαn 6= Bα

so X(∞).

Examples: flutes and ladders [DPS]

There exist hyperbolic manifolds Xwith rays α, β and αn → α

in each of the following situations:

(a) α � β and β � α but Bα 6= Bβ

(b) d∞(α, β) <∞ but Bα 6= Bβ

(c) d∞(α, β) =∞ but Bα = Bβ

(d) d∞(αn, αm) <∞ and Bαn = Bαm∀n,mbut d∞(αn, α) =∞ and Bαn 6= Bα

X non Hausdorff,(for any “visual” topology on ∂mX or ∂v X )

Compactifying by adding “horofunctions”X = general, complete Riemannian manifold

Xi↪→ C(X ) topological embedding

p 7→ d(p, ·)

p 7→ d(p, ·)X = i(X )

C(X)and ∂X = X − X

p 7→ −d(p, ·)X = i(X )

Xix↪→ C(X ) topological embedding

p 7→ d(x , p)− d(p, ·)X = ix(X )

p 7→ d(x , p)− d(p, ·)| {z }bp(x , ·) the horofunction cocycle

X = ix(X )C(X)

and ∂X = X − X

bp(x , y) = −bp(y , x)bp(x , y) + bp(y , z) = bp(x , z)

X = ix(X )C(X)

and ∂X = X − X

bp(x , ·)− bp(x ′, ·) = bp(x , x ′)

X = ix(X )C(X)

and ∂X = X − X

Xb↪→ C2(X ) = Ccocycle(X×X )

p 7→ bp(x , y) = xp − py| {z }horofunction cocycle

bp(x , ·)− bp(x ′, ·) = bp(x , x ′)

X = b(X )C2(X)

and ∂X = X − X

X = b(X )C2(X)

and ∂X = X − X

Gromov (horofunction) compactification of Xand Gromov (horofunction) boundary of X[M.Gromov, “Hyperbolic manifolds, groups and actions” (1978)]

a horofunction is a functionξ(x , y) = limpn→∞ bpn (x , y) ∈ ∂X

X = b(X )C2(X)

and ∂X = X − X

The Busemann functions are natural horofunctions:α ray

X = b(X )C2(X)

and ∂X = X − X

The Busemann functions are natural horofunctions:α ray Bα(x , y) = limt→∞ bα(t)(x , y) = limt→∞ xα(t)− α(t)y ∈ ∂X

X = b(X )C2(X)

and ∂X = X − X

B : R(X )→ ∂X the Busemann mapα 7→ Bα

X = b(X )C2(X)

and ∂X = X − X

∂BX .= B(R(X )) ⊂ ∂X the Busemann points of the boundary

1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence

The Busemann mapX = complete Riemannian manifold

p 7→ bp(x , y) = xp − py

X = b(X )C2(X)

∂X = X − X

p 7→ bp(x , y) = xp − py

X = b(X )C2(X)

∂X = X − X

Theorem. [Folklore: Gromov...]

Let X be a Cartan-Hadamard n-manifold:(i) the Busemann map B is continuous;(ii) B : Ro(X)→ ∂X is surjective + injective

⇒ X(∞) ∼= ∂X (∼= Sn−1)

p 7→ bp(x , y) = xp − py

X = b(X )C2(X)

∂X = X − X

⇒ X(∞) ∼= ∂X (∼= Sn−1)

Problem. Understand the map Bfor general (non simply-connected) manifolds:

1© is B continuous?2© when is B surjective?3© workable criteria for Bα = Bβ ?

(i.e for visual equivalence)

p 7→ bp(x , y) = xp − py

X = b(X )C2(X)

∂X = X − X

⇒ X(∞) ∼= ∂X (∼= Sn−1)

(i.e for visual equivalence)1©, 2©: NO, in general

p 7→ bp(x , y) = xp − py

X = b(X )C2(X)

∂X = X − X

⇒ X(∞) ∼= ∂X (∼= Sn−1)

(i.e for visual equivalence)1©, 2©: NO, in general (hyperbolic flutes and hyperbolic ladders)

Theorem. [DPS]

(i) B : R(X)→ ∂X is lower semi-continuous on any negatively curved manifold ;(ii) there exists a hyperbolic flute X and rays αn → α such that limn→∞ Bαn Bα.

Theorem. [DPS]

hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei

Theorem. [DPS]

Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group

gi hyperbolic/parabolic isometries of H2

Theorem. [DPS]

A(g±,o) = {x : d(x , o)≥d(x , g±1o)}the attractive/repulsive domains of g

Theorem. [DPS]

Schottky : A(g±1i , o)∩A(g±1

j , o) = ∅ ∀i 6= j

Theorem. [DPS]

j , o) = ∅ ∀i 6= j

If: • the axes gi ∩ gj = ∅• {A(g±i , o)} → ζ ∈ ∂H

Theorem. [DPS]

j , o) = ∅ ∀i 6= j

If: • the axes gi ∩ gj = ∅• {A(g±i , o)} → ζ ∈ ∂H

⇒ X = G\H2

hyperbolic fluteeach hyperbolic gi funneleach hyperbolic gi cusp

ζ the “infinite” end e

Theorem. [DPS]

j , o) = ∅ ∀i 6= j

For A(gi , o) and ζi as in the picture: αi =−→oζi , α =

−→oζ

αi = αi/G metrically asymptotic rays on X = G\H2

even: d∞(αi , αj ) = 0, so Bαi = Bαj

We have αi → α but: Bα 6= limi→∞ Bαi = Bα0

Twisted Hyperbolic flute

Theorem. [DPS]

There exists a hyperbolic ladder X → Σ2 with group G ∼= Z and rays α, α′ such that:(i) ∂BX consists of 4 points, while ∂X has a continuum of points;(ii) d∞(α, α′) <∞ and α ≺ α′ ≺ α, but Bα 6= Bα′ ;(iii) the limit set Gx0 ⊂ ∂X depends on x0, and for some x0 it is included in ∂X − ∂BX .

Theorem. [DPS]

hyperbolic ladder =

8<:Z-covering of a closed hyperbolic surface Σg of genus g ≥ 2obtained by glueing infinitely many copies of Σg −

Sgi=1 γi

with (γi ) simple, closed non-intersecting fundamental geodesics

Theorem. [DPS]

4 rays α, α−, α′, α′− metrically and visually non-asymptotic (as Bα 6= Bα− , Bα 6= Bα′ )

Theorem. [DPS]

4 rays α, α−, α′, α′− metrically and visually non-asymptotic (as Bα 6= Bα− , Bα 6= Bα′ )> Bα 6= Bα− obvious

Theorem. [DPS]

4 rays α, α−, α′, α′− metrically and visually non-asymptotic (as Bα 6= Bα− , Bα 6= Bα′ )> Bα 6= Bα− obvious

> Bα 6= Bα′ as (by direct computation) Bα(x, x′) > 0 Bα′ (x, x′) ′= Bα(x′, x) = −Bα(x, x′) < 0

Theorem. [DPS]

4 rays α, α−, α′, α′− metrically and visually non-asymptotic (as Bα 6= Bα− , Bα 6= Bα′ )> the hyperbolic metric every other ray is metrically strongly asymptotic (d∞ = 0)

to one of {α, α−, α′, α′−}⇒ ∂BX has 4 points

Theorem. [DPS]

a non-Busemann point: ξ = limi→∞ g i x0, for x0 in the middle

Theorem. [DPS]

> actually if bgi x0→ Bα (let’s say)

Theorem. [DPS]

> actually if bgi x0→ Bα (let’s say) ⇒ bgi x0

= b(gi x0)′ → Bα′ so Bα = Bα′ , contradiction.

X = G\H quotient of a Cartan-Hadamard manifold H, α ray on XMain idea: express Bα by Bα and the dynamics of G y H

horoball (through x): Hα(x)={y : Bα(x , y)≥0}horosphere (through x): ∂Hα(x)={y : Bα(x , y)=0}

the sup-level / level sets of Bα , containing x

Proposition. [DPS]

X =G\H quotient of a Cartan-Hadamard manifold H.Let α be a ray of X from o, α a lift to H from o:(i) ∀x ∈X ∃ a maximal horoball Hmax

α,x 63 Gx

(ii) Bα(o, x) = supg∈G Bα(o, gx) = ρ(o,Hmaxα,x )

Proposition. [DPS]

α,x 63 Gx

Criterium for visual equivalence. [DPS]

X =G\H quotient of a Cartan-Hadamard manifold H.– α, β rays of X with same origin o, α, β lifts from o– Hα(o),Hβ(o) horoballs of α, β through o:

(i) α � β iff ∃(gn) :

gnα+ → β+

g−1n o → Hα(o)

(ii) Bα = Bβ iff α � β and β � α

Proposition. [DPS]

α,x 63 Gx

Criterium for visual equivalence. [DPS]

X =G\H quotient of a Cartan-Hadamard manifold H.– α, β rays of X with same origin o, α, β lifts from o– Hα(o),Hβ(o) horoballs of α, β through o:

(i) α � β iff ∃(gn) :

gnα+ → β+

g−1n o → Hα(o)

(ii) Bα = Bβ iff α � β and β � α

for a Cartan-Hadamard manifold: ∂X = X(∞)

Bα = α(+∞) or α+

General resultsExample in dimension n = 3

Geometrically finite manifolds= a large class of (negatively curved) manifolds with finitely generated π1(X )

• dim(X) = 2 same as π1(X) f.g.• dim(X) > 2 stronger than π1(X) f.g.

X = G\H H = Cartan-Hadamard,−b2≤k(H)≤−a2< 0

LG the limit set of G, CG ⊂ H its convex hull

CX = G\CG ⊂ X the Nielsen core of X(the smallest closed and convex subset of X containing all thegeodesics which meet infinitely many often a compact set)

X is geometrically finite if some (any)ε-neighbourhood of CX has finite volume

CX = G\CG ⊂ X the Nielsen core of X

Theorem. [DPS]

Let X =G\H be a geometrically finite manifold, and α, β rays of X :(i) d∞(α, β) <∞ ⇔ α � β ⇔ Bα = Bβ(ii) B : R(X)→ ∂X is continuous and surjective ⇒ X(∞) ∼= R(X)/equiv. ∼= ∂X

if dim(X) = 2 X is a compact surface with boundaryif dim(X) > 2 X is a compact manifold with boundary

with a finite number of conical singularities(one for each conjugate class of maximal parabolic subgroups of G)

CX = G\CG ⊂ X the Nielsen core of X

Theorem. [DPS]

Let X =G\H be a geometrically finite manifold, and α, β rays of X :(i) d∞(α, β) <∞ ⇔ α � β ⇔ Bα = Bβ(ii) B : R(X)→ ∂X is continuous and surjective ⇒ X(∞) ∼= R(X)/equiv. ∼= ∂X

if dim(X) = 2 X is a compact surface with boundaryif dim(X) > 2 X is a compact manifold with boundary

with a finite number of conical singularities(one for each conjugate class of maximal parabolic subgroups of G)

ξ ∈ X is a conical singularity if it has a neighbourhood homeomorphic to the cone over some topological manifold)

The simplest (non-compact, non simply-connected) geometrically finite 3-manifold

P =<p> infinite cyclic parabolic group of H3, X = G\H2

LP = {ξ} the limit set Ord(P) = ∂H3 − ξ the discontinuity domainD(P, o) = {x ∈ H3 : d(x , o) ≤ d(x , pno), ∀n ∈ Z} the Dirichlet domain

1 see X = P\D(P, o) = P\[Hξ × (0,+∞)] ∼= Cil × (0,+∞)

2 add the ordinary Dirichlet points: X ′ = X ∪ [∂D(P, o)∩Ord(P)] ∼= Cil × [0,+∞)

3 adding one point corresponding to ξ ↔ P [with the topology: xn → ξ for any diverging (xn)]:X = X ′ ∪ {ξ} ∼= Cil × [0,+∞]/[B+=B−=(x,+∞)]

1 see X = P\D(P, o) = P\[Hξ × (0,+∞)] ∼= Cil × (0,+∞)

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