on the geometry of rays and the gromov compactification of ... › people › sambusetti › talks...
TRANSCRIPT
![Page 1: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/1.jpg)
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
![Page 2: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/2.jpg)
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
![Page 3: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/3.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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...
![Page 4: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/4.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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...
![Page 5: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/5.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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...
![Page 6: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/6.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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...
![Page 7: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/7.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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...
![Page 8: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/8.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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...
![Page 9: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/9.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”
Example. X = En
∂X = {half-lines}/parallelism ∼= Sn−1
X = X ∪ ∂X ∼= Bn−1
![Page 10: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/10.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”
Example. X = En
∂X = {half-lines}/parallelism ∼= Sn−1
X = X ∪ ∂X ∼= Bn−1
Näif idea: X general, complete Riemannian manifold
![Page 11: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/11.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”
Example. X = En
∂X = {half-lines}/parallelism ∼= Sn−1
X = X ∪ ∂X ∼= Bn−1
Näif idea: X general, complete Riemannian manifold
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
![Page 12: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/12.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”
Example. X = En
∂X = {half-lines}/parallelism ∼= Sn−1
X = X ∪ ∂X ∼= Bn−1
Näif idea: X general, complete Riemannian manifold
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)
![Page 13: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/13.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite 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–
![Page 14: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/14.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite 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
![Page 15: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/15.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite 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−−−−→bnα(tn)→ β for some {tn} → ∞, {bn} → β(0)
−→xy = a minimizing geodesic segment from x to y
Technical fact: the correct definition of visual convergenceasks for a sequence bn → β(0) with
−−−−→bnα(tn)→ β
![Page 16: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/16.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite 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−−−−→bnα(tn)→ β for some {tn} → ∞, {bn} → β(0)
−→xy = a minimizing geodesic segment from x to y
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!
![Page 17: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/17.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite 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−−−−→bnα(tn)→ β for some {tn} → ∞, {bn} → β(0)
(not symmetric, apriori)
![Page 18: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/18.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”Parallelism for rays on a general, complete Riemannian manifold X :
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
![Page 19: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/19.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”Parallelism for rays on a general, complete Riemannian manifold X :
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
![Page 20: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/20.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”Parallelism for rays on a general, complete Riemannian manifold X :
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)
![Page 21: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/21.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”Parallelism for rays on a general, complete Riemannian manifold X :
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”
![Page 22: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/22.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “directions”Parallelism for rays on a general, complete Riemannian manifold X :
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”
Bα = the Busemann function of the ray α
![Page 23: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/23.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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)
![Page 24: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/24.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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)
Fact: always converges
![Page 25: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/25.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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)
Fact: always converges
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)
![Page 26: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/26.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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)
Fact: always converges
Folklore: X = any Riemannian manifold
Bα(x , y) = d(x , y) iff (−→xy is a ray and) α � −→xy .
![Page 27: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/27.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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)
Fact: always converges
Folklore: X = any Riemannian manifold
Bα(x , y) = d(x , y) iff (−→xy is a ray and) α � −→xy .
⇓if Bα = Bβ then α � γ iff β � γ for every γ (and reciprocally)
![Page 28: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/28.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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)
Fact: always converges
Folklore: X = any Riemannian manifold
Bα(x , y) = d(x , y) iff (−→xy is a ray and) α � −→xy .
⇓if Bα = Bβ then α � γ iff β � γ for every γ, which implies
Theorem. [Folklore: Busemann, Shihoama et al.]
Two rays α and β are visually asymptotic from every point o iff Bα = Bβ .
![Page 29: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/29.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
![Page 30: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/30.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
![Page 31: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/31.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
![Page 32: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/32.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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? R(X) has the uniform topology (u.c. on compacts)which means: αn → α iff α′n(0)→ α′(0).
![Page 33: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/33.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
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).
![Page 34: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/34.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
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.
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).
![Page 35: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/35.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
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.
Dropping the “simply connected” assumption?
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
![Page 36: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/36.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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?
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.
Dropping the “simply connected” assumption?
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 )
![Page 37: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/37.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
![Page 38: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/38.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xi↪→ C(X ) topological embedding
p 7→ d(p, ·)
![Page 39: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/39.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xi↪→ C(X ) topological embedding
p 7→ d(p, ·)X = i(X )
C(X)and ∂X = X − X
![Page 40: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/40.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xi↪→ C(X ) topological embedding
p 7→ −d(p, ·)X = i(X )
C(X)and ∂X = X − X
![Page 41: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/41.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xix↪→ C(X ) topological embedding
p 7→ d(x , p)− d(p, ·)X = ix(X )
C(X)and ∂X = X − X
![Page 42: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/42.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xix↪→ C(X ) topological embedding
p 7→ d(x , p)− d(p, ·)| {z }bp(x , ·) the horofunction cocycle
X = ix(X )C(X)
and ∂X = X − X
![Page 43: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/43.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xix↪→ C(X ) topological embedding
p 7→ d(x , p)− d(p, ·)| {z }bp(x , ·) the horofunction cocycle
bp(x , y) = −bp(y , x)bp(x , y) + bp(y , z) = bp(x , z)
X = ix(X )C(X)
and ∂X = X − X
![Page 44: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/44.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xix↪→ C(X ) topological embedding
p 7→ d(x , p)− d(p, ·)| {z }bp(x , ·) the horofunction cocycle
bp(x , y) = −bp(y , x)bp(x , y) + bp(y , z) = bp(x , z)
bp(x , ·)− bp(x ′, ·) = bp(x , x ′)
X = ix(X )C(X)
and ∂X = X − X
![Page 45: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/45.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py| {z }horofunction cocycle
bp(x , y) = −bp(y , x)bp(x , y) + bp(y , z) = bp(x , z)
bp(x , ·)− bp(x ′, ·) = bp(x , x ′)
X = b(X )C2(X)
and ∂X = X − X
![Page 46: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/46.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py| {z }horofunction cocycle
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
![Page 47: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/47.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py| {z }horofunction cocycle
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
The Busemann functions are natural horofunctions:α ray
![Page 48: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/48.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py| {z }horofunction cocycle
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
The Busemann functions are natural horofunctions:α ray Bα(x , y) = limt→∞ bα(t)(x , y) = limt→∞ xα(t)− α(t)y ∈ ∂X
![Page 49: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/49.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py| {z }horofunction cocycle
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
The Busemann functions are natural horofunctions:α ray Bα(x , y) = limt→∞ bα(t)(x , y) = limt→∞ xα(t)− α(t)y ∈ ∂X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
![Page 50: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/50.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
Compactifying by adding “horofunctions”X = general, complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py| {z }horofunction cocycle
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
The Busemann functions are natural horofunctions:α ray Bα(x , y) = limt→∞ bα(t)(x , y) = limt→∞ xα(t)− α(t)y ∈ ∂X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
∂BX .= B(R(X )) ⊂ ∂X the Busemann points of the boundary
![Page 51: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/51.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
The Busemann mapX = complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py
X = b(X )C2(X)
∂X = X − X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
![Page 52: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/52.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
The Busemann mapX = complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py
X = b(X )C2(X)
∂X = X − X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
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)
![Page 53: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/53.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
The Busemann mapX = complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py
X = b(X )C2(X)
∂X = X − X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
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)
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)
![Page 54: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/54.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
The Busemann mapX = complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py
X = b(X )C2(X)
∂X = X − X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
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)
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)1©, 2©: NO, in general
![Page 55: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/55.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
The Busemann mapX = complete Riemannian manifold
Xb↪→ C2(X ) = Ccocycle(X×X )
p 7→ bp(x , y) = xp − py
X = b(X )C2(X)
∂X = X − X
B : R(X )→ ∂X the Busemann mapα 7→ Bα
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)
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)1©, 2©: NO, in general (hyperbolic flutes and hyperbolic ladders)
![Page 56: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/56.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
![Page 57: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/57.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
![Page 58: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/58.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group
gi hyperbolic/parabolic isometries of H2
![Page 59: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/59.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group
gi hyperbolic/parabolic isometries of H2
A(g±,o) = {x : d(x , o)≥d(x , g±1o)}the attractive/repulsive domains of g
![Page 60: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/60.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group
gi hyperbolic/parabolic isometries of H2
Schottky : A(g±1i , o)∩A(g±1
j , o) = ∅ ∀i 6= j
![Page 61: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/61.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group
gi hyperbolic/parabolic isometries of H2
Schottky : A(g±1i , o)∩A(g±1
j , o) = ∅ ∀i 6= j
If: • the axes gi ∩ gj = ∅• {A(g±i , o)} → ζ ∈ ∂H
![Page 62: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/62.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group
gi hyperbolic/parabolic isometries of H2
Schottky : A(g±1i , o)∩A(g±1
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
![Page 63: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/63.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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.
hyperbolic flute =(topologically) S2 with infinitely many punctures ei → limit puncture e;(geometrically) complete hyp. structure with cusps/funnels ∀ei
Construction of hyperbolic flutes:G =<g1,..., gk ,...>∞-generated Schottky group
gi hyperbolic/parabolic isometries of H2
Schottky : A(g±1i , o)∩A(g±1
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
![Page 64: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/64.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
![Page 65: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/65.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
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
![Page 66: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/66.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
![Page 67: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/67.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
![Page 68: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/68.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
4 rays α, α−, α′, α′− metrically and visually non-asymptotic (as Bα 6= Bα− , Bα 6= Bα′ )
![Page 69: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/69.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
4 rays α, α−, α′, α′− metrically and visually non-asymptotic (as Bα 6= Bα− , Bα 6= Bα′ )> Bα 6= Bα− obvious
![Page 70: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/70.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
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
![Page 71: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/71.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
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
![Page 72: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/72.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
a non-Busemann point: ξ = limi→∞ g i x0, for x0 in the middle
![Page 73: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/73.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
a non-Busemann point: ξ = limi→∞ g i x0, for x0 in the middle
> actually if bgi x0→ Bα (let’s say)
![Page 74: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/74.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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 .
a non-Busemann point: ξ = limi→∞ g i x0, for x0 in the middle
> actually if bgi x0→ Bα (let’s say) ⇒ bgi x0
= b(gi x0)′ → Bα′ so Bα = Bα′ , contradiction.
![Page 75: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/75.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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
![Page 76: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/76.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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
![Page 77: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/77.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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
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 )
![Page 78: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/78.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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
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 )
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
![Page 79: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/79.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
1© Continuity of the Busemann map2© Surjectivity of the Busemann map3© Busemann (visual) equivalence
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
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 )
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+
![Page 80: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/80.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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.
![Page 81: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/81.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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
![Page 82: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/82.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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
X is geometrically finite if some (any)ε-neighbourhood of CX has finite volume
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)
![Page 83: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/83.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
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
X is geometrically finite if some (any)ε-neighbourhood of CX has finite volume
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)
![Page 84: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/84.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]
![Page 85: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/85.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]
![Page 86: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/86.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]
![Page 87: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/87.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]
![Page 88: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/88.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]
![Page 89: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/89.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]
![Page 90: On the Geometry of Rays and the Gromov Compactification of ... › people › sambusetti › talks › talklux.pdfOn the Geometry of Rays and the Gromov Compactification of Negatively](https://reader033.vdocuments.mx/reader033/viewer/2022053015/5f156ab5743d823f977df1a3/html5/thumbnails/90.jpg)
History“Näives” Compactifications
The Gromov CompactificationThe Busemann map
Geometrically finite manifolds
General resultsExample in dimension n = 3
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,+∞)]