The Structure Of Properly Convex Manifolds - Florida State University
The Structure of Properly Convex ManifoldsSam Ballas(joint with D. Long)University of PittsburghMay 14, 2014
Some Questions What are convex projective manifolds?
Some Questions What are convex projective manifolds? How are they similar to hyperbolic manifolds? How arethey different?
Some Questions What are convex projective manifolds? How are they similar to hyperbolic manifolds? How arethey different? What sort of structure do convex projective manifoldshave?
Some Questions What are convex projective manifolds?Generalizations of Hyperbolic manifolds How are they similar to hyperbolic manifolds? How arethey different? What sort of structure do convex projective manifoldshave?
Some Questions What are convex projective manifolds?Generalizations of Hyperbolic manifolds How are they similar to hyperbolic manifolds? How arethey different?Strictly Convex very similar. Properly convex lesssimilar What sort of structure do convex projective manifoldshave?
Some Questions What are convex projective manifolds?Generalizations of Hyperbolic manifolds How are they similar to hyperbolic manifolds? How arethey different?Strictly Convex very similar. Properly convex lesssimilar What sort of structure do convex projective manifoldshave?Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
Projective Space RP n is the space of lines through origin in Rn 1 .
Projective Space RP n is the space of lines through origin in Rn 1 . Let P : Rn 1 \{0} RP n be the obvious projection.
Projective Space RP n is the space of lines through origin in Rn 1 . Let P : Rn 1 \{0} RP n be the obvious projection. The automorphism group of RP n isPGLn 1 (R) : GLn 1 (R)/R .
Projective Space RP n is the space of lines through origin in Rn 1 . Let P : Rn 1 \{0} RP n be the obvious projection. The automorphism group of RP n isPGLn 1 (R) : GLn 1 (R)/R . A codimension k projective plane is the projectivization ofa codimension k plane in Rn 1
Projective Space RP n is the space of lines through origin in Rn 1 . Let P : Rn 1 \{0} RP n be the obvious projection. The automorphism group of RP n isPGLn 1 (R) : GLn 1 (R)/R . A codimension k projective plane is the projectivization ofa codimension k plane in Rn 1 A projective line is the projectivization of a 2-plane in Rn 1
Projective Space RP n is the space of lines through origin in Rn 1 . Let P : Rn 1 \{0} RP n be the obvious projection. The automorphism group of RP n isPGLn 1 (R) : GLn 1 (R)/R . A codimension k projective plane is the projectivization ofa codimension k plane in Rn 1 A projective line is the projectivization of a 2-plane in Rn 1 A projective hyperplane is the projectivization of an n-planein Rn 1 .
A Decomposition of RP n Let H be a hyperplane in Rn 1 . H gives rise to a Decomposition of RP n Rn t RP n 1 intoan affine part and an ideal part.
A Decomposition of RP n Let H be a hyperplane in Rn 1 . H gives rise to a Decomposition of RP n Rn t RP n 1 intoan affine part and an ideal part.
A Decomposition of RP n Let H be a hyperplane in Rn 1 . H gives rise to a Decomposition of RP n Rn t RP n 1 intoan affine part and an ideal part. RP n \P(H) is called an affine patch.
What is convex projective geometry?Motivation from hyperbolic geometry Let hx, y i x1 y1 . . . xn yn xn 1 yn 1 be the standardbilinear form of signature (n, 1) on Rn 1 Let C {x Rn 1 hx, xi 0}
What is convex projective geometry?Motivation from hyperbolic geometry Let hx, y i x1 y1 . . . xn yn xn 1 yn 1 be the standardbilinear form of signature (n, 1) on Rn 1 Let C {x Rn 1 hx, xi 0} P(C) is the Klein model of hyperbolic space. P(C) has isometry group PSO(n, 1) PGLn 1 (R)
What is convex projective geometry?Motivation from hyperbolic geometryNice Properties of Hyperbolic Space Convex: Intersection with projective lines is connected.
What is convex projective geometry?Motivation from hyperbolic geometryNice Properties of Hyperbolic Space Convex: Intersection with projective lines is connected. Properly Convex: Convex and closure is contained in anaffine patch Disjoint from some projective hyperplane.
What is convex projective geometry?Motivation from hyperbolic geometryNice Properties of Hyperbolic Space Convex: Intersection with projective lines is connected. Properly Convex: Convex and closure is contained in anaffine patch Disjoint from some projective hyperplane. Strictly Convex: Properly convex and boundary containsno non-trivial projective line segments.
What is convex projective geometry?Motivation from hyperbolic geometryConvex projective geometry focuses on the geometry ofmanifolds that are locally modeled on properly (strictly) convexdomains.
What is convex projective geometry?Motivation from hyperbolic geometryConvex projective geometry focuses on the geometry ofmanifolds that are locally modeled on properly (strictly) convexdomains.Hyperbolic GeometryHn /ΓΓ Isom(Hn )Γ discrete torsion free
What is convex projective geometry?Motivation from hyperbolic geometryConvex projective geometry focuses on the geometry ofmanifolds that are locally modeled on properly (strictly) convexdomains.Hyperbolic GeometryHn /ΓΓ Isom(Hn )Γ discrete torsion freeConvex ProjectiveGeometryΩ/ΓΩ properly (strictly) convexΓ PGL(Ω)Γ discrete torsion free
What is Convex Projective GeometryExamples1. Hyperbolic manifolds
What is Convex Projective GeometryExamples1. Hyperbolic manifolds2. Let T be the interior of a triangle in RP 2 and let Γ Diag be a suitable lattice inside the group of 3 3 diagonalmatrices with determinant 1 and distinct positiveeigenvalues. T /Γ is a properly convex torus.
What is Convex Projective GeometryExamples1. Hyperbolic manifolds2. Let T be the interior of a triangle in RP 2 and let Γ Diag be a suitable lattice inside the group of 3 3 diagonalmatrices with determinant 1 and distinct positiveeigenvalues. T /Γ is a properly convex torus.These are extreme examples of properly convex manifolds.Generic examples interpolate between these extreme cases.
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.Every properly convex set Ω admits a Hilbert metric given by x b y a dΩ (x, y ) log[a, x; y , b] log x a y b
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.Every properly convex set Ω admits a Hilbert metric given by x b y a dΩ (x, y ) log[a, x; y , b] log x a y b When Ω is an ellipsoid dΩ is twice the hyperbolic metric.
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.Every properly convex set Ω admits a Hilbert metric given by x b y a dΩ (x, y ) log[a, x; y , b] log x a y b When Ω is an ellipsoid dΩ is twice the hyperbolic metric. PGL(Ω) Isom(Ω) and equal when Ω is strictly convex.
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.Every properly convex set Ω admits a Hilbert metric given by x b y a dΩ (x, y ) log[a, x; y , b] log x a y b When Ω is an ellipsoid dΩ is twice the hyperbolic metric. PGL(Ω) Isom(Ω) and equal when Ω is strictly convex. Discrete subgroups of PGL(Ω) act properly discontinuouslyon Ω.
Classification of Isometriesa la Cooper, Long, TillmannIf Ω is open and properly convex then PGL(Ω) embeds inSL n 1 (R) which allows us to talk about eigenvalues.
Classification of Isometriesa la Cooper, Long, TillmannIf Ω is open and properly convex then PGL(Ω) embeds inSL n 1 (R) which allows us to talk about eigenvalues.If γ PGL(Ω) then γ is1. elliptic if γ fixes a point in Ω (zero translation length realized),2. parabolic if γ acts freely on Ω and has all eigenvalues ofmodulus 1 (zero translation length not realized), and3. hyperbolic otherwise (positive translation length)
Similarities to Hyperbolic IsometriesStrictly Convex Case1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.
Similarities to Hyperbolic IsometriesStrictly Convex Case1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.2. When Ω is strictly convex, parabolic isometries have aunique fixed point on Ω.
Similarities to Hyperbolic IsometriesStrictly Convex Case1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.2. When Ω is strictly convex, parabolic isometries have aunique fixed point on Ω.3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on Ω and act by translation along the lineconnecting them.
Similarities to Hyperbolic IsometriesStrictly Convex Case1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.2. When Ω is strictly convex, parabolic isometries have aunique fixed point on Ω.3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on Ω and act by translation along the lineconnecting them.4. In particular, when Ω is strictly convex, hyperbolicisometries are positive proximal (eigenvalues of minimumand maximum modulus are unique, real, and positive)
Similarities to Hyperbolic IsometriesThe General CaseA properly convex domain is a compact convex subset of Rnand so if γ PGL(Ω) then Brouwer fixed point theorem applies
Similarities to Hyperbolic IsometriesThe General CaseA properly convex domain is a compact convex subset of Rnand so if γ PGL(Ω) then Brouwer fixed point theorem applies Elliptic elements are all conjugate into O(n).
Similarities to Hyperbolic IsometriesThe General CaseA properly convex domain is a compact convex subset of Rnand so if γ PGL(Ω) then Brouwer fixed point theorem applies Elliptic elements are all conjugate into O(n). Parabolic elements have a connected fixed set in Ω.
Similarities to Hyperbolic IsometriesThe General CaseA properly convex domain is a compact convex subset of Rnand so if γ PGL(Ω) then Brouwer fixed point theorem applies Elliptic elements are all conjugate into O(n). Parabolic elements have a connected fixed set in Ω. Hyperbolic elements have an attracting and repellingsubspaces A and A in Ω. The action on these sets isorthogonal and their dimension is determined by thenumber of “powerful” Jordan blocks of γ
Margulis LemmaLet Ω RP n is an open properly convex domain and letΓ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x Ω then the groupΓx hγ Γ dΩ (x, γx) µn iis virtually nilpotent.
Margulis LemmaLet Ω RP n is an open properly convex domain and letΓ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x Ω then the groupΓx hγ Γ dΩ (x, γx) µn iis virtually nilpotent. Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x].
Margulis LemmaLet Ω RP n is an open properly convex domain and letΓ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x Ω then the groupΓx hγ Γ dΩ (x, γx) µn iis virtually nilpotent. Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x]. The Margulis lemma places restrictions on the topologyand geometry of the “thin” part of Ω/Γ.
Margulis LemmaLet Ω RP n is an open properly convex domain and letΓ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x Ω then the groupΓx hγ Γ dΩ (x, γx) µn iis virtually nilpotent. Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x]. The Margulis lemma places restrictions on the topologyand geometry of the “thin” part of Ω/Γ.Result due to Gromov-Margulis-Thurston for Hn andCooper-Long-Tillmann in general.
Rigidity and FlexibilityWhen n 3 Mostow-Prasad rigidity tells us that complete finitevolume hyperbolic structures are very rigidTheorem 1 (Mostow ’70, Prasad ’73)Let n 3 and suppose that Hn /Γ1 and Hn /Γ2 both have finitevolume. If Γ1 and Γ2 are isomorphic then Hn /Γ1 and Hn /Γ2 areisometric.
Rigidity and FlexibilityWhen n 3 Mostow-Prasad rigidity tells us that complete finitevolume hyperbolic structures are very rigidTheorem 1 (Mostow ’70, Prasad ’73)Let n 3 and suppose that Hn /Γ1 and Hn /Γ2 both have finitevolume. If Γ1 and Γ2 are isomorphic then Hn /Γ1 and Hn /Γ2 areisometric.There is no Mostow-Prasad rigidity for properly (strictly) convexdomains.There are examples of finite volume hyperbolic manifoldswhose complete hyperbolic structure can be “deformed” to anon-hyperbolic convex projective structure.
Deformations Start with M0 Ω0 /Γ0 which is properly convex.
Deformations Start with M0 Ω0 /Γ0 which is properly convex. “Perturb” Γ0 to Γ1 PGL(Ω1 ) PGLn 1 (R), where Γ0 Γ1and Ω1 is properly convex.
Deformations Start with M0 Ω0 /Γ0 which is properly convex. “Perturb” Γ0 to Γ1 PGL(Ω1 ) PGLn 1 (R), where Γ0 Γ1and Ω1 is properly convex. We say that M1 Ω1 /Γ1 is a deformation of M0
Deformations Start with M0 Ω0 /Γ0 which is properly convex. “Perturb” Γ0 to Γ1 PGL(Ω1 ) PGLn 1 (R), where Γ0 Γ1and Ω1 is properly convex. We say that M1 Ω1 /Γ1 is a deformation of M0Ex: Let Ω0 Hn , Γ0 PSO(n, 1), such that Ω0 /Γ0 is finitevolume and contains an embedded totally geodesichypersurface Σ. Let Γ1 be obtained by “bending” along Σ.
Structure of Hyperbolic ManifoldsThe Closed CaseLet Hn /Γ be a closed hyperbolic manifold. Since Γ acts cocompactly by isometries on Hn we see thatΓ is δ-hyperbolic group (Švarc-Milnor)
Structure of Hyperbolic ManifoldsThe Closed CaseLet Hn /Γ be a closed hyperbolic manifold. Since Γ acts cocompactly by isometries on Hn we see thatΓ is δ-hyperbolic group (Švarc-Milnor) By compactness, we see that if 1 6 γ Γ then γ ishyperbolic
Structure of Hyperbolic ManifoldsThe Closed CaseLet Hn /Γ be a closed hyperbolic manifold. Since Γ acts cocompactly by isometries on Hn we see thatΓ is δ-hyperbolic group (Švarc-Milnor) By compactness, we see that if 1 6 γ Γ then γ ishyperbolic In particular, if 1 6 γ Γ then γ is positive proximal
Structure of Convex Projective ManifoldsThe Closed CaseLet M Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 Ω0 /Γ0 .
Structure of Convex Projective ManifoldsThe Closed CaseLet M Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 Ω0 /Γ0 .Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic.
Structure of Convex Projective ManifoldsThe Closed CaseLet M Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 Ω0 /Γ0 .Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic. Proof sketch.If Ω is not strictly convex then it willcontain arbitrarily fat triangles and isthus not δ-hyperbolic.
Structure of Convex Projective ManifoldsThe Closed CaseLet M Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 Ω0 /Γ0 .Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic. Proof sketch.If Ω is not strictly convex then it willcontain arbitrarily fat triangles and isthus not δ-hyperbolic. Since Γ actscocompactly by isometries on Ω,Švarc-Milnor tells us that Ω is q.i. to Γand is thus δ-hyperbolic.
Structure of Convex Projective ManifoldsThe Closed CaseTheorem 3 (Benoist)Let 1 6 γ Γ then γ is positive proximal.Proof. Again by compactness we have that if 1 6 γ Γ then γ ishyperbolic.
Structure of Convex Projective ManifoldsThe Closed CaseTheorem 3 (Benoist)Let 1 6 γ Γ then γ is positive proximal.Proof. Again by compactness we have that if 1 6 γ Γ then γ ishyperbolic. Since Ω is strictly convex and γ is hyperbolic we see that γhas exactly 2 fixed points in Ω and acts as translationalong the geodesic connecting them. γ is thus positiveproximal.
Structure of Hyperbolic ManifoldsFinite Volume CaseLet M Hn /Γ be a finite volume hyperbolic manifold. We candecompose M asGM MKCi ,iwhere MK is a compact and π1 (MK ) Γ and Ci arecomponents of the thin part called cusps.
Structure of Hyperbolic ManifoldsFinite Volume CaseLet M Hn /Γ be a finite volume hyperbolic manifold. We candecompose M asGM MKCi ,iwhere MK is a compact and π1 (MK ) Γ and Ci arecomponents of the thin part called cusps.As we will see, the Margulis lemma tells us that the Ci haverelatively simple geometry.
Geometry of the CuspsLet C be a cusp of a finite volume hyperbolic manifold and let 1 v T v 2 n 1 P v R0 In 1 v 001be the group of parabolic translations fixing . Let x0 Hn ,then C B/ where B is horoball bounded by Px0 and is afinite extension of a lattice in P.
Structure of Hyperbolic ManifoldsThe Finite Volume Case Γ no longer acts cocompactly on Hn and Γ is no longerδ-hyperbolic
Structure of Hyperbolic ManifoldsThe Finite Volume Case Γ no longer acts cocompactly on Hn and Γ is no longerδ-hyperbolic Instead Γ is δ-hyperbolic relative to the cusps
Structure of Hyperbolic ManifoldsThe Finite Volume Case Γ no longer acts cocompactly on Hn and Γ is no longerδ-hyperbolic Instead Γ is δ-hyperbolic relative to the cusps If 1 6 γ Γ is freely homotopic into a cusp then γ isparabolic, otherwise γ is hyperbolic (positive proximal)
Structures of Convex Projective ManifoldsThe Strictly Convex Finite Volume CaseLet Ω/Γ be a finite volume (Hausdorff measure of Hilbertmetric) strictly convex manifold.Theorem 4 (Cooper, Long, Tillmann ‘11)Let M Ω/Γ be as above thenF M MK i Ci , where MK is compact and Ci is projectivelyequivalent to the cusp of a finite volume hyperbolicmanifold, Γ is δ-hyperbolic relative to its cusps, and If 1 6 γ Γ is freely homotopic into a cusp then γ isparabolic. Otherwise γ is hyperbolic (positive proximal).
Figure-8 ExampleConsider the following example.Let K be the figure-8 knot, let M S 3 \K , and let G π1 (M)
Figure-8 ExampleConsider the following example.Let K be the figure-8 knot, let M S 3 \K , and let G π1 (M)Theorem 5 (B)There exists ε 0 such that for each t ( ε, ε) there is aproperly convex domain Ωt and a discrete group Γt PGL(Ωt )such that Ωt /Γt M, Ω0 /Γ0 is the complete hyperbolic structure on M, and If t 6 0 then Ωt is not strictly convex.
Figure-8 ExampleTheorem 6 (B)For each t ( ε, ε) we can decompose Ωt /Γt as MKtwhere MKt is compact and C t T 2 [1, ).FCt , For each t, C t Bt / t , where t is a lattice an Abeliangroup Pt of “translations,” and Bt is a “horoball” bounded byan orbit of Pt .
Figure-8 Example
Figure-8 Example
Figure-8 Example
Figure-8 Example
Figure-8 Example For each t 6 0 there is 1 6 γt Γt such that γt ishyperbolic, freely homotopic into C t , but not positiveproximal.
Figure-8 Example For each t 6 0 there is 1 6 γt Γt such that γt ishyperbolic, freely homotopic into C t , but not positiveproximal. Ωt contains non-trivial line segments in Ωt that arepreserved by conjugates of t . In particular, Ωt is notδ-hyperbolic.
Figure-8 ExampleTheorem 7 (B, Long)1 6 γ Γt is positive proximal if and only if it cannot be freelyhomotoped into C t .
Figure-8 ExampleTheorem 7 (B, Long)1 6 γ Γt is positive proximal if and only if it cannot be freelyhomotoped into C t .Proof. Let 1 6 γ Γt . No elements of Pt are positive proximal, so ifγ is freely homotopic to C t then it is not positive proximal.
Figure-8 ExampleTheorem 7 (B, Long)1 6 γ Γt is positive proximal if and only if it cannot be freelyhomotoped into C t .Proof. Let 1 6 γ Γt . No elements of Pt are positive proximal, so ifγ is freely homotopic to C t then it is not positive proximal. If γ is not freely homotopic to C t then γ has positivetranslation length and is thus hyperbolic. Furthermore, thistranslation length is realized by points on an axis.
Figure-8 ExampleProof (Continued).Use Margulis lemma to construct a disjoint and Γt invariantcollection Ht of horoballs in Ωt .
Figure-8 ExampleProof (Continued).Use Margulis lemma to construct a disjoint and Γt invariantcollection Ht of horoballs in Ωt .let Ω̂t be the electric space obtained by collapsing thehorospherical boundary components of Ωt \Ht .
Figure-8 ExampleProof (Continued).Use Margulis lemma to construct a disjoint and Γt invariantcollection Ht of horoballs in Ωt .let Ω̂t be the electric space obtained by collapsing thehorospherical boundary components of Ωt \Ht .Lemma 8 (B, Long)Ω̂t is δ-hyperbolic
Figure-8 ExampleProof (Continued). Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign.
Figure-8 ExampleProof (Continued). Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign. If γ is not positive proximal then there will be a γ-invariantset T Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary
Figure-8 ExampleProof (Continued). Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign. If γ is not positive proximal then there will be a γ-invariantset T Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary
Figure-8 ExampleProof (Continued). Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign. If γ is not positive proximal then there will be a γ-invariantset T Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary This gives rise to arbitrarily fat triangles in Ω̂t
Summary and Questions The structure of a finite volume strictly convex manifold iswell understood.
Summary and Questions The structure of a finite volume strictly convex manifold iswell understood. As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
Summary and Questions The structure of a finite volume strictly convex manifold iswell understood. As you deform the structure the “coarse” geometry of thecompact part doesn’t change. The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
Summary and Questions The structure of a finite volume strictly convex manifold iswell understood. As you deform the structure the “coarse” geometry of thecompact part doesn’t change. The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma. Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
Summary and Questions The structure of a finite volume strictly convex manifold iswell understood. As you deform the structure the “coarse” geometry of thecompact part doesn’t change. The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma. Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3 Theorem 7 should hold for higher dimensions.
Summary and Questions The structure of a finite volume strictly convex manifold iswell understood. As you deform the structure the “coarse” geometry of thecompact part doesn’t change. The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma. Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3 Theorem 7 should hold for higher dimensions. What can we say for deformations of deformations ofinfinite volume hyperbolic manifolds?
What is convex projective geometry? Motivation from hyperbolic geometry Nice Properties of Hyperbolic Space Convex: Intersection with projective lines is connected. Properly Convex: Convex and closure is contained in an affine patch ()Disjoint from some projective hyperplane. Strictly Convex: Properly convex and boundary contains
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