The convex real projective manifolds and orbifolds with radial ends I: the openness of deformations (Preliminary)

Transcription

1 1/46 The convex real projective manifolds and orbifolds with radial ends I: the openness of deformations (Preliminary) Suhyoung Choi Department of Mathematical Science KAIST, Daejeon, South Korea mathsci.kaist.ac.kr/ schoi (Copies of my lectures are posted)

2 2/46 Abstract A real projective orbifold is an n-dimensional orbifold modeled on RP n with the group PGL(n + 1, R). We concentrate on an orbifold with a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to (n 1)-dimensional orbifolds times intervals.

3 2/46 Abstract A real projective orbifold is an n-dimensional orbifold modeled on RP n with the group PGL(n + 1, R). We concentrate on an orbifold with a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to (n 1)-dimensional orbifolds times intervals. A real projective orbifold has radial ends if a neighborhood of each end is foliated by projective geodesics concurrent to one another. It is said to be convex if any path can be homotoped to a projective geodesic with endpoints fixed.

4 2/46 Abstract A real projective orbifold is an n-dimensional orbifold modeled on RP n with the group PGL(n + 1, R). We concentrate on an orbifold with a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to (n 1)-dimensional orbifolds times intervals. A real projective orbifold has radial ends if a neighborhood of each end is foliated by projective geodesics concurrent to one another. It is said to be convex if any path can be homotoped to a projective geodesic with endpoints fixed. A real projective structure on such an orbifold sometimes admits deformations to inequivalent parameters of real projective structures, giving us a nontrivial deformation space.

5 3/46 Abstract continued We will prove the local homeomorphism between the deformation space of real projective structures on such an orbifold with radial ends with various conditions and the PGL(n + 1, R)-representation space of the fundamental group with corresponding conditions. Here, we have to restrict each end to have a fundamental group isomorphic to a finite extension of a product of hyperbolic groups and abelian groups.

6 3/46 Abstract continued We will prove the local homeomorphism between the deformation space of real projective structures on such an orbifold with radial ends with various conditions and the PGL(n + 1, R)-representation space of the fundamental group with corresponding conditions. Here, we have to restrict each end to have a fundamental group isomorphic to a finite extension of a product of hyperbolic groups and abelian groups. We will use a Hessian argument to show that a small deformation of a real projective orbifold with ends will remain properly and strictly convex in a generalized sense if so is the beginning real projective orbifold, provided that the ends behave in a convex manner. We will also prove the closedness of the convex real projective structures on orbifolds with irreducibilty condition.

7 4/46 Remarks The understanding of the radial ends is not accomplished in this paper as this forms an another subject. We assume certain "natural" conditions to ends general enough and amenable to study. These will be our base colonies. We are trying to classify all possible types of ends (with Yves Benoist.) Even simple examples are unclear.

8 4/46 Remarks The understanding of the radial ends is not accomplished in this paper as this forms an another subject. We assume certain "natural" conditions to ends general enough and amenable to study. These will be our base colonies. We are trying to classify all possible types of ends (with Yves Benoist.) Even simple examples are unclear. This work is mostly theoretical, and we need more computable examples. What might be the good conditions on the ends for computational purposes? (Benoist) (For Coxeter orbifolds, see Choi and Marquis.) I gave computations talk in Melbourne Univ. (in my homepage). This work is still in progress. Consulting experts still..

9 4/46 Remarks The understanding of the radial ends is not accomplished in this paper as this forms an another subject. We assume certain "natural" conditions to ends general enough and amenable to study. These will be our base colonies. We are trying to classify all possible types of ends (with Yves Benoist.) Even simple examples are unclear. This work is mostly theoretical, and we need more computable examples. What might be the good conditions on the ends for computational purposes? (Benoist) (For Coxeter orbifolds, see Choi and Marquis.) I gave computations talk in Melbourne Univ. (in my homepage). This work is still in progress. Consulting experts still.. Some of this might be known also by Cooper and Tillmann independently and simultaneously: More computed examples for 3-manifolds admitting complete hyperbolic structures.

10 5/46 Outline Introduction Orbifolds and real projective structures Deformation spaces and holonomy maps Classification of ends: rather restrictions on ends Main results: Openness Convexity and convex domains The IPC-structures and relative hyperbolicity The proof of the main results Main Examples Classification of the ends

11 6/46 Introduction Orbifolds and real projective structures Orbifolds By an n-dimensional orbifold, we mean a Hausdorff second countable topological space with a fine open cover {U i, i I} with models (Ũi, G i ) where G i is a finite group acting on the open subset Ũi of R n and a map p i : Ũi U i inducing homeomorphism Ũi /G i U i where for each i, j, x U i U j, there exists U k with x U k U i U j. given an inclusion U j U i induces an equivariant map Ũj Ũi with respect to G j G i.

12 6/46 Introduction Orbifolds and real projective structures Orbifolds By an n-dimensional orbifold, we mean a Hausdorff second countable topological space with a fine open cover {U i, i I} with models (Ũi, G i ) where G i is a finite group acting on the open subset Ũi of R n and a map p i : Ũi U i inducing homeomorphism Ũi /G i U i where for each i, j, x U i U j, there exists U k with x U k U i U j. given an inclusion U j U i induces an equivariant map Ũj Ũi with respect to G j G i. A real projective structure on an orbifold is given by having charts from U i s to open subsets of RP n with transition maps in PGL(n + 1, R).

13 6/46 Introduction Orbifolds and real projective structures Orbifolds By an n-dimensional orbifold, we mean a Hausdorff second countable topological space with a fine open cover {U i, i I} with models (Ũi, G i ) where G i is a finite group acting on the open subset Ũi of R n and a map p i : Ũi U i inducing homeomorphism Ũi /G i U i where for each i, j, x U i U j, there exists U k with x U k U i U j. given an inclusion U j U i induces an equivariant map Ũj Ũi with respect to G j G i. A real projective structure on an orbifold is given by having charts from U i s to open subsets of RP n with transition maps in PGL(n + 1, R). A good orbifold: M/Γ where Γ is a discrete group with a properly discontinuous order.

14 7/46 Introduction Orbifolds and real projective structures Real projective structures on orbifolds Suppose that a discrete group Γ act on a manifold M properly discontinuously. A real projective structure on M/Γ with simply connected M is given by an immersion D : M RP n equivariant with respect to a homomorphism h : Γ PGL(n + 1, R) where Γ is the fundamental group of M/Γ.

15 7/46 Introduction Orbifolds and real projective structures Real projective structures on orbifolds Suppose that a discrete group Γ act on a manifold M properly discontinuously. A real projective structure on M/Γ with simply connected M is given by an immersion D : M RP n equivariant with respect to a homomorphism h : Γ PGL(n + 1, R) where Γ is the fundamental group of M/Γ. A real projective structure on M/Γ is convex if D(M) is a convex domain in an affine subspace A n RP n. In this case, we will identify M with D(M) for a particular choice of D and Γ with its image under h. A properly convex domain is a convex domain that is a precompact domain in some affine subspace. A convex domain is properly convex iff it does not contain a complete real line.

16 7/46 Introduction Orbifolds and real projective structures Real projective structures on orbifolds Suppose that a discrete group Γ act on a manifold M properly discontinuously. A real projective structure on M/Γ with simply connected M is given by an immersion D : M RP n equivariant with respect to a homomorphism h : Γ PGL(n + 1, R) where Γ is the fundamental group of M/Γ. A real projective structure on M/Γ is convex if D(M) is a convex domain in an affine subspace A n RP n. In this case, we will identify M with D(M) for a particular choice of D and Γ with its image under h. A properly convex domain is a convex domain that is a precompact domain in some affine subspace. A convex domain is properly convex iff it does not contain a complete real line. A real projective structure on M/Γ is properly convex if so is D(M).

17 8/46 Introduction Orbifolds and real projective structures Figure: The developing images of 2-orbifolds

18 9/46 Introduction Orbifolds and real projective structures Motivations to study real projective structures The study of lattices are very much established with many techniques. Flexible geometric structures parametrize representations in many cases and they do not correspond to lattice situations mostly.

19 9/46 Introduction Orbifolds and real projective structures Motivations to study real projective structures The study of lattices are very much established with many techniques. Flexible geometric structures parametrize representations in many cases and they do not correspond to lattice situations mostly. Real projective and conformal structures are often the most flexible finite-dimensional types we can study. Other geometries are subgeometries. Orbifolds with convex real projective structures have many properties of CAT (0)-spaces and the theory is compatible with the geometric group theory. (no angles... )

20 9/46 Introduction Orbifolds and real projective structures Motivations to study real projective structures The study of lattices are very much established with many techniques. Flexible geometric structures parametrize representations in many cases and they do not correspond to lattice situations mostly. Real projective and conformal structures are often the most flexible finite-dimensional types we can study. Other geometries are subgeometries. Orbifolds with convex real projective structures have many properties of CAT (0)-spaces and the theory is compatible with the geometric group theory. (no angles... ) There are "much" more orbifolds with real projective structures than homogeneous Riemannian ones.

21 10/46 Introduction Orbifolds and real projective structures Motivations to study real projective structures on 3-orbifolds Real projective structures on surfaces and 2-orbifolds are understood". There is a constructive classification from the convex decomposition theorem and the annulus decomposition theorem. There is a study by Cooper, Long, and Thistleswaite on the real projective structures obtained by deforming hyperbolic 3-manifolds in the Hodgson-Weeks census of About 5 percents are deformable.

22 10/46 Introduction Orbifolds and real projective structures Motivations to study real projective structures on 3-orbifolds Real projective structures on surfaces and 2-orbifolds are understood". There is a constructive classification from the convex decomposition theorem and the annulus decomposition theorem. There is a study by Cooper, Long, and Thistleswaite on the real projective structures obtained by deforming hyperbolic 3-manifolds in the Hodgson-Weeks census of About 5 percents are deformable. For the reflection 3-orbifolds, a study of orderable reflection orbifolds by Choi and Marquis. For more complicated reflection 3-orbifolds, Choi, Hodgson, and Lee made some studies on hyperbolic cubes and dodecahedra. (Some deform as bending and as nonbendings and some do not.) Ideal or hyperideal reflection 3-orbifolds without order two deform always.

23 10/46 Introduction Orbifolds and real projective structures Motivations to study real projective structures on 3-orbifolds Real projective structures on surfaces and 2-orbifolds are understood". There is a constructive classification from the convex decomposition theorem and the annulus decomposition theorem. There is a study by Cooper, Long, and Thistleswaite on the real projective structures obtained by deforming hyperbolic 3-manifolds in the Hodgson-Weeks census of About 5 percents are deformable. For the reflection 3-orbifolds, a study of orderable reflection orbifolds by Choi and Marquis. For more complicated reflection 3-orbifolds, Choi, Hodgson, and Lee made some studies on hyperbolic cubes and dodecahedra. (Some deform as bending and as nonbendings and some do not.) Ideal or hyperideal reflection 3-orbifolds without order two deform always. Question (Cooper): Does every hyperbolic 3-orbifold deform up to finite covers? How to make sense?

24 11/46 Introduction Deformation spaces and holonomy maps Deformation spaces of convex real projective structures Given an orbifold S, a convex real projective structure is given by a diffeomorphism f : S Ω/Γ for a convex domain Ω in RP n and Γ a subgroup of PGL(n + 1, R). This induces a diffeomorphism D : S Ω equivariant with respect to h : π 1 (S) Γ.

25 11/46 Introduction Deformation spaces and holonomy maps Deformation spaces of convex real projective structures Given an orbifold S, a convex real projective structure is given by a diffeomorphism f : S Ω/Γ for a convex domain Ω in RP n and Γ a subgroup of PGL(n + 1, R). This induces a diffeomorphism D : S Ω equivariant with respect to h : π 1 (S) Γ. The deformation space CDef(S) of convex real projective structures is {(D, h)}/ where (D, h) (D, h ) if there is an isotopy f : S S such that D = D f and h ( f g f 1 ) = h(g) for each g π 1 (S) or D = k D and h ( ) = kh( )k 1 for k PGL(n + 1, R).

26 11/46 Introduction Deformation spaces and holonomy maps Deformation spaces of convex real projective structures Given an orbifold S, a convex real projective structure is given by a diffeomorphism f : S Ω/Γ for a convex domain Ω in RP n and Γ a subgroup of PGL(n + 1, R). This induces a diffeomorphism D : S Ω equivariant with respect to h : π 1 (S) Γ. The deformation space CDef(S) of convex real projective structures is {(D, h)}/ where (D, h) (D, h ) if there is an isotopy f : S S such that D = D f and h ( f g f 1 ) = h(g) for each g π 1 (S) or D = k D and h ( ) = kh( )k 1 for k PGL(n + 1, R). Alternatively, CDef(S) = {f : S Ω/Γ}/ where f g for f : S Ω/Γ and g : S Ω /Γ if there exists a projective diffeomorphism k : Ω/Γ Ω /Γ so that k f is homotopic to g.

27 12/46 Introduction Deformation spaces and holonomy maps End orbifold A real projective orbifold has radial ends if each end has an end neighborhood foliated by concurrent geodesics for each chart ending at the common point of concurrency. Each end has a neighborhood diffeomorphic to a closed orbifold times an open interval.

28 12/46 Introduction Deformation spaces and holonomy maps End orbifold A real projective orbifold has radial ends if each end has an end neighborhood foliated by concurrent geodesics for each chart ending at the common point of concurrency. Each end has a neighborhood diffeomorphic to a closed orbifold times an open interval. Given an end, there is an end orbifold associated with the end. The radial foliation has a transversal real projective structure and hence the end orbifold has an induced real projective structure of one dimension lower. The end orbifold is convex if O is convex. If the end orbifold is properly convex, then we say that the end is a transversely properly convex end.

29 13/46 Introduction Deformation spaces and holonomy maps The reason why we study orbifolds with ends The orbifolds with ends might be more computable. For example, ideal or hyperideal reflection hyperbolic 3-orbifolds with no edge order two have local deformation space of dimension 6.

30 13/46 Introduction Deformation spaces and holonomy maps The reason why we study orbifolds with ends The orbifolds with ends might be more computable. For example, ideal or hyperideal reflection hyperbolic 3-orbifolds with no edge order two have local deformation space of dimension 6. The general type of ends are not understood. The theory should be at least as complicated as that of Kleinian groups.

31 13/46 Introduction Deformation spaces and holonomy maps The reason why we study orbifolds with ends The orbifolds with ends might be more computable. For example, ideal or hyperideal reflection hyperbolic 3-orbifolds with no edge order two have local deformation space of dimension 6. The general type of ends are not understood. The theory should be at least as complicated as that of Kleinian groups. We (Jaejeong Lee, a student of Kapovich) are exploring some examples currently. Also, Crampon and Marquis are studying these theoretically with horospherical ends from the generalized Kleinian group perspectives. Radial properly convex ones have some chance of classifications (with Benoist).

32 14/46 Introduction Deformation spaces and holonomy maps We define Def E (O) to be the subspace of Def(O) consisting of real projective structures with radial ends. The representation space rep(π 1 (O), PGL(n + 1, R)) is the quotient space of the homomorphism space Hom(π 1 (O), PGL(n + 1, R))/PGL(n + 1, R) where PGL(n + 1, R) acts by conjugation h( ) gh( )g 1.

33 14/46 Introduction Deformation spaces and holonomy maps We define Def E (O) to be the subspace of Def(O) consisting of real projective structures with radial ends. The representation space rep(π 1 (O), PGL(n + 1, R)) is the quotient space of the homomorphism space Hom(π 1 (O), PGL(n + 1, R))/PGL(n + 1, R) where PGL(n + 1, R) acts by conjugation h( ) gh( )g 1. We define rep E (π 1 (O), PGL(n + 1, R)) to be the subspace of representations where each end fundamental group has a nonzero common eigenvector. The end fundamental group condition: If a representation of an end fixes a point of RP n, then it fixes a unique one.

34 15/46 Introduction Deformation spaces and holonomy maps The hol map: The local homeomorphism property Theorem A Let O be a noncompact topologically tame n-orbifold. Suppose that O has the end fundamental group conditions. Then the following map is a local homeomorphism: hol : Def E (O) rep E (π 1 (O), PGL(n + 1, R)). Proof. This follows as in the compact cases using the bump functions. However, we may need the bump functions extending to the ends for radial ends.

35 16/46 Introduction Classification of ends: rather restrictions on ends A subdomain K of RP n is said to be horospherical if it is strictly convex and the boundary K is diffeomorphic to R n 1 and bdk K is a single point. K is lens-shaped if it is a convex domain and K is a disjoint union of two smoothly embedded (n 1)-cells not containing any straight segment in them. A cone is a domain in RP n whose closure in RP n has a point in the boundary, called a cone-point, so that every other point has a segment contained in the domain with endpoint the cone point and itself.

36 16/46 Introduction Classification of ends: rather restrictions on ends A subdomain K of RP n is said to be horospherical if it is strictly convex and the boundary K is diffeomorphic to R n 1 and bdk K is a single point. K is lens-shaped if it is a convex domain and K is a disjoint union of two smoothly embedded (n 1)-cells not containing any straight segment in them. A cone is a domain in RP n whose closure in RP n has a point in the boundary, called a cone-point, so that every other point has a segment contained in the domain with endpoint the cone point and itself. A cone-over a lens-shaped domain A is a convex submanifold that contains a lens-shaped domain A of the same dimension and is a union of segments from a cone-point v A to points of A, the manifold boundary is one of the two boundary components of A, and each maximal segment from v meets the two boundary components at unique points.

37 17/46 Introduction Classification of ends: rather restrictions on ends Figure: The universal covers of horospherical and lens shaped ends. The radial lines form cone-structures.

38 17/46 Introduction Classification of ends: rather restrictions on ends A lens-cone is the union of the segments over a lens-shaped domain. A lens is the lens-shaped domain A, not determined uniquely by the lens-cone itself.

39 17/46 Introduction Classification of ends: rather restrictions on ends A lens-cone is the union of the segments over a lens-shaped domain. A lens is the lens-shaped domain A, not determined uniquely by the lens-cone itself. A totally-geodesic subdomain is a convex domain in a hyperspace. A cone-over a totally-geodesic domain A is a cone over a point x not in the hyperspace.

40 17/46 Introduction Classification of ends: rather restrictions on ends A lens-cone is the union of the segments over a lens-shaped domain. A lens is the lens-shaped domain A, not determined uniquely by the lens-cone itself. A totally-geodesic subdomain is a convex domain in a hyperspace. A cone-over a totally-geodesic domain A is a cone over a point x not in the hyperspace. In general, a join of two convex sets C 1 and C 2 is the union of segments with end points in C 1 and C 2 respectively and is denoted by C 1 + C 2 in this paper. We can generalize to the sum of n sets C 1,..., C n. A cone-over a joined domain is a one containing a joined domain A and is a union of segments from a cone-point A to points of A where the cone point is given by V V.

41 18/46 Introduction Classification of ends: rather restrictions on ends If every subgroup of finite index of a group Γ has a finite center, Γ is said to be virtual center-free group or a vcf-group. An admissible group is a finite extension of a finite product of Z l Γ 1 Γ k

42 18/46 Introduction Classification of ends: rather restrictions on ends If every subgroup of finite index of a group Γ has a finite center, Γ is said to be virtual center-free group or a vcf-group. An admissible group is a finite extension of a finite product of Z l Γ 1 Γ k Let E be an (n 1)-dimensional end orbifold, and let µ be a holonomy representation π 1 (E) PGL(n + 1, R) fixing a point x. Suppose that µ(π 1 (E)) acts on a lens-shaped domain K in RP n not containing x with boundary a union of two open (n 1)-cells A and B and π 1 (E) acts properly on A and B with compact Hausdorff quotients and the cone of K over x exists. Then µ is said to be a lens-shaped representation for E with respect to x.

43 18/46 Introduction Classification of ends: rather restrictions on ends If every subgroup of finite index of a group Γ has a finite center, Γ is said to be virtual center-free group or a vcf-group. An admissible group is a finite extension of a finite product of Z l Γ 1 Γ k Let E be an (n 1)-dimensional end orbifold, and let µ be a holonomy representation π 1 (E) PGL(n + 1, R) fixing a point x. Suppose that µ(π 1 (E)) acts on a lens-shaped domain K in RP n not containing x with boundary a union of two open (n 1)-cells A and B and π 1 (E) acts properly on A and B with compact Hausdorff quotients and the cone of K over x exists. Then µ is said to be a lens-shaped representation for E with respect to x. µ is a totally-geodesic representation if µ(π 1 (E) acts on a cone-over a totally-geodesic subdomain with a cone-point x. If µ(π 1 (E)) acts on a horospherical domain K, then µ is said to be a horospherical representation. In this case, bdk K = {x}. If µ(π 1 (E)) acts on a joined domain, its cone-point and associated subspaces V i s and V j s, then µ is said to be a joined representation.

44 19/46 Introduction Classification of ends: rather restrictions on ends We say that the radial end is admissible if either it has a neighborhood whose universal cover is a horospherical domain or is a cone over a lens-shaped domain for the corresponding representation of π 1 (E) for a corresponding end orbifold E. We require that the cone-point has to correspond to the end of the radial lines for the given radial end.

45 19/46 Introduction Classification of ends: rather restrictions on ends We say that the radial end is admissible if either it has a neighborhood whose universal cover is a horospherical domain or is a cone over a lens-shaped domain for the corresponding representation of π 1 (E) for a corresponding end orbifold E. We require that the cone-point has to correspond to the end of the radial lines for the given radial end. We will also say that an admissible end is hyperbolic if the end fundamental group is hyperbolic and is Benoist if k = l 1. Benoist or hyperbolic ends are said to be permanantly properly convex. l k follows from the result of Benoist.

46 19/46 Introduction Classification of ends: rather restrictions on ends We say that the radial end is admissible if either it has a neighborhood whose universal cover is a horospherical domain or is a cone over a lens-shaped domain for the corresponding representation of π 1 (E) for a corresponding end orbifold E. We require that the cone-point has to correspond to the end of the radial lines for the given radial end. We will also say that an admissible end is hyperbolic if the end fundamental group is hyperbolic and is Benoist if k = l 1. Benoist or hyperbolic ends are said to be permanantly properly convex. l k follows from the result of Benoist. We have k = 1 and l = 0 if and only if the end fundamental group is hyperbolic. There are hyperbolic lens ends that are not totally geodesic. We can bend... Lens and l k 1 imply totally geodesic

47 20/46 Introduction Classification of ends: rather restrictions on ends Convex end fundamental condition Loosely speaking, this condition is one where if a representation of an end of convex orbifolds with radial admissible ends fixes a point of RP n, then it fixes a unique one. This is more general than a mere end fundamental condition. For example, the end has a 1-dimensional singularity following the radial line, this is true.

48 20/46 Introduction Classification of ends: rather restrictions on ends Convex end fundamental condition Loosely speaking, this condition is one where if a representation of an end of convex orbifolds with radial admissible ends fixes a point of RP n, then it fixes a unique one. This is more general than a mere end fundamental condition. For example, the end has a 1-dimensional singularity following the radial line, this is true. More generally if the fundamental group is virtually center free, this is true. (Irreducibility essentially proves this..)

49 20/46 Introduction Classification of ends: rather restrictions on ends Convex end fundamental condition Loosely speaking, this condition is one where if a representation of an end of convex orbifolds with radial admissible ends fixes a point of RP n, then it fixes a unique one. This is more general than a mere end fundamental condition. For example, the end has a 1-dimensional singularity following the radial line, this is true. More generally if the fundamental group is virtually center free, this is true. (Irreducibility essentially proves this..) For abelian fundamental groups, we need some singularity as above...

50 21/46 Introduction Classification of ends: rather restrictions on ends Let O be a tame n-orbifold where end fundamental groups are admissible. We define rep E,ce (π 1 (O), PGL(n + 1, R)) to be the subspace of rep E (π 1 (O), PGL(n + 1, R)) where each end is realized as admissible end of some real projective orbifold mapping into O as an end. (The realization is essentially unique.)

51 21/46 Introduction Classification of ends: rather restrictions on ends Let O be a tame n-orbifold where end fundamental groups are admissible. We define rep E,ce (π 1 (O), PGL(n + 1, R)) to be the subspace of rep E (π 1 (O), PGL(n + 1, R)) where each end is realized as admissible end of some real projective orbifold mapping into O as an end. (The realization is essentially unique.) We define rep i E,ce (π 1(O), PGL(n + 1, R)) as... We define Def i E,ce (O) to be the deformation space of real projective structures with admissible ends and irreducible holonomy and define CDef E,ce (O) to be the deformation space of irreducible properly convex-structures with admissible ends (or IPC-structures).

52 22/46 Introduction Classification of ends: rather restrictions on ends Theorem B Let O be a noncompact topologically tame n-orbifold with admissible ends. Suppose that O satisfies the convex end fundamental group conditions. Then In Def i E,ce (O), the subspace CDef E (O) of IPC-structures is open. Suppose further that π 1 (O) contains no notrivial nilpotent normal subgroup. The deformation space CDef E,ce (O) of IPC-structures on O maps homeomorphic to a component of rep i E,ce (π 1(O), PGL(n + 1, R)).

53 23/46 Introduction Classification of ends: rather restrictions on ends Theorem C Suppose that O is IPC. If every straight arc in the boundary of the domain Õ is contained in the closure of a component of a chosen equivarient set of end neighborhoods in Õ, then O is said to be strictly convex with respect to the collection of the ends. And O is also said to have a strict IPC-structure with respect to the collection of the ends.

54 23/46 Introduction Classification of ends: rather restrictions on ends Theorem C Suppose that O is IPC. If every straight arc in the boundary of the domain Õ is contained in the closure of a component of a chosen equivarient set of end neighborhoods in Õ, then O is said to be strictly convex with respect to the collection of the ends. And O is also said to have a strict IPC-structure with respect to the collection of the ends. We show that an IPC-orbifold O with admissible end is strictly IPC iff π 1 (O) is relatively hyperbolic with respect to its end fundamental groups using Bowditch and Drutu-Sapir s work.

55 24/46 Introduction Classification of ends: rather restrictions on ends Theorem C Theorem C Let O be a strict IPC noncompact topologically tame n-dimensional orbifold with admissible ends and convex end fundamental group condition. Suppose also that O has no essential homotopy annulus or torus. Then π 1 (O) is relatively hyperbolic with respect to its end fundamental groups.

56 24/46 Introduction Classification of ends: rather restrictions on ends Theorem C Theorem C Let O be a strict IPC noncompact topologically tame n-dimensional orbifold with admissible ends and convex end fundamental group condition. Suppose also that O has no essential homotopy annulus or torus. Then π 1 (O) is relatively hyperbolic with respect to its end fundamental groups. In Def i E,ce (O), the subspace SDefi E (O) of strict IPC-structures with respect to the ends is open. The deformation space SDef E,ce (O) of strict IPC-structures on O with respect to the ends maps homeomorphic to a component of rep i E,ce (π 1(O), PGL(n + 1, R)).

57 25/46 Convexity and convex domains Convexity. We begin by discussing the convexity: Proposition (Vey) A real projective orbifold with nonempty radial end is convex if and only if the developing map sends the universal cover to a convex open domain in RP n. A real projective orbifold with nonempty radial end is properly convex if and only if the developing map sends the universal cover to a properly convex open domain in a compact domain in an affine patch of RP n. If a convex real projective orbifold with nonempty radial end is not properly convex, then its holonomy is reducible.

58 26/46 Convexity and convex domains Proposition (Benoist) Suppose that a discrete subgroup Γ of PGL(n, R) acts on a properly convex (n 1)-dimensional open domain Ω so that Ω/Γ is compact. Then the following statements are equivalent. Every subgroup of finite index of Γ has a finite center. Every subgroup of finite index of Γ has a trivial center. Every subgroup of finite index of Γ is irreducible in PGL(n, R). That is, Γ is strongly irreducible. The Zariski closure of Γ is semisimple. Γ does not contain a normal infinite nilpotent subgroup. Γ does not contain a normal infinite abelian subgroup. The group with property (i) above is said to be the group with trivial virtual center.

59 27/46 Convexity and convex domains Theorem (Benoist) Let Γ be a discrete subgroup of PGL(n, R) with a trivial virtual center. Suppose that a discrete subgroup Γ of PGL(n, R) acts on a properly convex (n 1)-dimensional open domain Ω so that Ω/Γ is compact. Then every representation of a component of Hom(Γ, PGL(n, R)) containing the inclusion representation also acts on a properly convex (n 1)-dimensional open domain cocompactly.

60 27/46 Convexity and convex domains Theorem (Benoist) Let Γ be a discrete subgroup of PGL(n, R) with a trivial virtual center. Suppose that a discrete subgroup Γ of PGL(n, R) acts on a properly convex (n 1)-dimensional open domain Ω so that Ω/Γ is compact. Then every representation of a component of Hom(Γ, PGL(n, R)) containing the inclusion representation also acts on a properly convex (n 1)-dimensional open domain cocompactly. Remark: This is the theorem we wish to generalize in the setting.

61 the Auslander conjecture.) 28/46 Convexity and convex domains Facts on horospherical ends Proposition Let O be a topologically tame properly convex real projective n-orbifold with radial ends. For each horospherical end, the space of ray from the end point form a complete affine space of dimension n 1. The only eigenvalues of g for an element of a horospherical end fundamental group are 1 or complex numbers of absolute value 1. An end point of a horospherical end cannot be on a segment in bdõ. For any compact set K inside a horospherical neighborhood, there exists a horospherical ellipsoid neighborhood disjoint from K. Let E be a complete end. Suppose that π 1 (E) has holonomy with eigenvalues of absolute value 1 only. Then E is horospherical. Comment: Actually, I don t need the eigenvalue condition for the final item. (Related to

62 29/46 Convexity and convex domains If π 1 (E) is hyperbolic, then E is lens. But if π 1 (E) has many factors, this is unclear.

63 29/46 Convexity and convex domains If π 1 (E) is hyperbolic, then E is lens. But if π 1 (E) has many factors, this is unclear. Proposition Suppose that M is a topologically tame properly convex real projective orbifold with radial ends with admissible end fundamental groups. Assume that M is not covered by a real line times a compact (n 1)-orbifold. Suppose that each end fundamental group is generated by closed curves about singularities or has the holonomy fixing the end vertex with eigenvalues 1. If each end is either horospherical or has a compact totally geodesic properly convex hyperspace in end neighborhoods, then the ends are admissible.

64 29/46 Convexity and convex domains If π 1 (E) is hyperbolic, then E is lens. But if π 1 (E) has many factors, this is unclear. Proposition Suppose that M is a topologically tame properly convex real projective orbifold with radial ends with admissible end fundamental groups. Assume that M is not covered by a real line times a compact (n 1)-orbifold. Suppose that each end fundamental group is generated by closed curves about singularities or has the holonomy fixing the end vertex with eigenvalues 1. If each end is either horospherical or has a compact totally geodesic properly convex hyperspace in end neighborhoods, then the ends are admissible. Let O be a 3-orbifold with the end orbifolds S3,3,3 2. Then the orbifold has admissible ends.

65 30/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Reminding Definitions Again Definition We will only study irreducible properly convex real projective structures on O, i.e., properly convex structures with irreducible holonomy representations and convex ends. We also need a condition that a straight arc in the boundary of Õ must be contained in the closure of some end neighborhood of an end-vertex and as a consequence any triangle with interior in Õ and boundary in bdõ must be inside an end-neighborhood. We call these two conditions no edge condition. The IPC-structure satisfying the no edge condition is said to be the strict IPC-structures.

66 31/46 Convexity and convex domains The IPC-structures and relative hyperbolicity A Hilbert metric on an IPC-structure is defined as a distance metric given by cross ratios. (We do not assume strictness here.) Let Ω be a properly convex domain. Then d Ω (p, q) = log(o, s, q, p) where o and s are endpoints of the maximal segment in Ω containing p, q.

67 31/46 Convexity and convex domains The IPC-structures and relative hyperbolicity A Hilbert metric on an IPC-structure is defined as a distance metric given by cross ratios. (We do not assume strictness here.) Let Ω be a properly convex domain. Then d Ω (p, q) = log(o, s, q, p) where o and s are endpoints of the maximal segment in Ω containing p, q. This gives us a well-defined Finsler metric. Given an IPC-structure on O, there is a Hilbert metric d H on Õ and hence on Õ. This induces a metric on O.

68 32/46 Convexity and convex domains The IPC-structures and relative hyperbolicity We will use Bowditch s result to show Theorem (D) Let O be a topologically tame strictly IPC-orbifold with radial ends and has no essential annuli or tori. Then π 1 (O) is relatively hyperbolic with respect to the end groups π 1 (E 1 ),..., π 1 (E k ).

69 32/46 Convexity and convex domains The IPC-structures and relative hyperbolicity We will use Bowditch s result to show Theorem (D) Let O be a topologically tame strictly IPC-orbifold with radial ends and has no essential annuli or tori. Then π 1 (O) is relatively hyperbolic with respect to the end groups π 1 (E 1 ),..., π 1 (E k ). Fact: If π 1 (E l ),.., π 1 (E k ) are hyperbolic for some 0 l < k, then π 1 (O) is relatively hyperbolic with respect to π 1 (E 1 ),..., π 1 (E l 1 ). (Drutu)

70 33/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof: We denote this quotient space bdõ 1 / by B. We will use Theorem 0.1. of Yaman [6]: We show that π 1 (O) acts on the set B as a geometrically finite convergence group.

71 33/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof: We denote this quotient space bdõ 1 / by B. We will use Theorem 0.1. of Yaman [6]: We show that π 1 (O) acts on the set B as a geometrically finite convergence group. The group acts properly discontinuously on the set of triples in B.

72 33/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof: We denote this quotient space bdõ 1 / by B. We will use Theorem 0.1. of Yaman [6]: We show that π 1 (O) acts on the set B as a geometrically finite convergence group. The group acts properly discontinuously on the set of triples in B. An end group Γ x for end vertex x is a parabolic subgroup fixing x since the elements in Γ x fixes only the contracted set in B and since there are no essential annuli. The groups of form Γ x are the only parabolic subgroups. Also, (B {x})/γ x is easily seen to be homeomorphic to the end orbifold and therefore, compact. Hence, each Γ x is a bounded parabolic subgroup.

73 34/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof continued: Let p be a point that is not a horospherical endpoint or a singleton corresponding an lens-shaped end. We show that p is a conical limit point.

74 34/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof continued: Let p be a point that is not a horospherical endpoint or a singleton corresponding an lens-shaped end. We show that p is a conical limit point. We find a sequence of holonomy transformations γ i and distinct points a, b X so that γ i (p) a and γ i (q) b for all q X {p}. To do this, we draw a line l(t) from a point of the fundamental domain to p where as t, l(t) p in the compactification.

75 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof continued: Let p be a point that is not a horospherical endpoint or a singleton corresponding an lens-shaped end. We show that p is a conical limit point. We find a sequence of holonomy transformations γ i and distinct points a, b X so that γ i (p) a and γ i (q) b for all q X {p}. To do this, we draw a line l(t) from a point of the fundamental domain to p where as t, l(t) p in the compactification. p' l q' m p q Figure: A shortest geodesic m to a geodesic l. 34/46

76 35/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Converse We will prove the partial converse to the above Theorem D: Theorem (E) Let O be a topologically tame IPC-orbifold with admissible ends without essential annuli or tori. Suppose that π 1 (O) is relatively hyperbolic group with respect to the admissible end groups π 1 (E 1 ),..., π 1 (E k ) where E i are horospherical for i = 1,..., m and lens-shaped for i = m + 1,..., k for 0 m k. Assume that O is IPC. Then O is strictly IPC. Let O 1 be obtained by removing the concave neighborhoods of hyperbolic ends. Then if O is IPC, then O 1 is strictly IPC. (Note: This improves the theorem in the paper.)

77 36/46 Convexity and convex domains The IPC-structures and relative hyperbolicity Proof. Suppose not. We obtain a triangle T with T in Õ 1. Lemma Suppose that O is a topologically tame properly convex n-orbifold with radial ends that are properly convex or horospherical and π 1 (O) is relatively hyperbolic with respect to its ends. O has no essential tori or essential annulus. Then every triangle T in Õ with T Õ is contained in the closure of a convex hull of its end. Proof. Uses asymptotic cone in Drutu-Sapir s work.

78 37/46 Convexity and convex domains The proof of the main results Then we define SDef E,ce (O) to be the subspace of Def i E,ce (O) consisting of strict IPC-structures. Also, we have SDef RP n,e,ce(o) Def i RP n,e,ce (O).

79 37/46 Convexity and convex domains The proof of the main results Then we define SDef E,ce (O) to be the subspace of Def i E,ce (O) consisting of strict IPC-structures. Also, we have SDef RP n,e,ce(o) Def i RP n,e,ce (O). Theorem Let O be a topologically tame real projective n-orbifold with admissible ends. Suppose that O satisfies the end fundamental group condition or more generally the convex end fundamental group conditions, and suppose that O has no essential homotopy annulus. In Def i E,ce (O), the subspace CDef E (O, ce) of IPC-structures is open, and so is SDef E (O, ce).

80 38/46 Convexity and convex domains The proof of the main results Corollary Let O be a topologically tame real projective n-orbifold with admissible ends. Suppose that O satisfies the end fundamental group condition or more generally the convex end fundamental group conditions, and suppose that O has no essential homotopy annulus. hol : CDef E,ce (O) rep E,ce (π 1 (O), PGL(n + 1, R)) is a local homeomorphism. Furthermore, if O has a strict IPC-structure with admissible ends, then so is hol : SDef E,ce (O) rep E,ce (π 1 (O), PGL(n + 1, R)).

81 39/46 Convexity and convex domains The proof of the main results Proof We increase the end neighborhoods to approximate Õ. In the affine suspension cone V and its dual cone V, we find Koszul-Vinberg function f V (x) = e φ(x) dφ (1) V

82 39/46 Convexity and convex domains The proof of the main results Proof We increase the end neighborhoods to approximate Õ. In the affine suspension cone V and its dual cone V, we find Koszul-Vinberg function f V (x) = e φ(x) dφ (1) V Then, we deform O. We patch together the deformed functions to obtain a function with positive definite Hessian. This implies the convexity. This proves the openess of IPC-structures.

83 39/46 Convexity and convex domains The proof of the main results Proof We increase the end neighborhoods to approximate Õ. In the affine suspension cone V and its dual cone V, we find Koszul-Vinberg function f V (x) = e φ(x) dφ (1) V Then, we deform O. We patch together the deformed functions to obtain a function with positive definite Hessian. This implies the convexity. This proves the openess of IPC-structures. To show the openness of strict IPC-structures, we need the fact that small deformation of O preserves relative hyperbolicity and hence the strictness of the IPC-structures.

84 40/46 Convexity and convex domains The proof of the main results Closedness Theorem Let O be a topologically tame IPC n-dimensional orbifold with convex end fundamental group condition. Assume that π 1 (O) has no nontrivial nilpotent normal subgroup. Then the following hold: The deformation space CDef E,ce (O) of IPC-structures on O maps homeomorphic to the union of components of rep i E,ce (π 1(O), PGL(n + 1, R)).

85 40/46 Convexity and convex domains The proof of the main results Closedness Theorem Let O be a topologically tame IPC n-dimensional orbifold with convex end fundamental group condition. Assume that π 1 (O) has no nontrivial nilpotent normal subgroup. Then the following hold: The deformation space CDef E,ce (O) of IPC-structures on O maps homeomorphic to the union of components of rep i E,ce (π 1(O), PGL(n + 1, R)). Suppose also that O has no essential homotopy annulus or torus. Similarly, the same can be said for SDef E,ce (O) of strict IPC-structures on O.

86 40/46 Convexity and convex domains The proof of the main results Closedness Theorem Let O be a topologically tame IPC n-dimensional orbifold with convex end fundamental group condition. Assume that π 1 (O) has no nontrivial nilpotent normal subgroup. Then the following hold: The deformation space CDef E,ce (O) of IPC-structures on O maps homeomorphic to the union of components of rep i E,ce (π 1(O), PGL(n + 1, R)). Suppose also that O has no essential homotopy annulus or torus. Similarly, the same can be said for SDef E,ce (O) of strict IPC-structures on O. We can drop the irreducibility in the representations space: Corollary Assume as above: If the ends of O are permanently properly convex, then hol maps the deformation space of IPC-structures on O homeomorphic to a union of components of rep E,ce (π 1 (O), PGL(n + 1, R)).

87 41/46 Convexity and convex domains Main Examples S. Tillman s example There is a census of small hyperbolic orbifolds with graph-singularity [4]. (See the paper by D. Heard, C. Hodgson, B. Martelli, and C. Petronio) There is a complete hyperbolic structure on the orbifold based on S 3 with handcuff singularity with two points removed. The singularity orders are three.

88 41/46 Convexity and convex domains Main Examples S. Tillman s example There is a census of small hyperbolic orbifolds with graph-singularity [4]. (See the paper by D. Heard, C. Hodgson, B. Martelli, and C. Petronio) There is a complete hyperbolic structure on the orbifold based on S 3 with handcuff singularity with two points removed. The singularity orders are three. There is a one-parameter space of deformations of the structures to real projective structures by simple matrix computations. These are all stricly IPC by our theory.

89 42/46 Convexity and convex domains Main Examples Main examples Theorem Suppose that a 3-dimensional orbifold is triangulated into one or two tetrahedra with edges in the singular locus and the vertices are all removed. Suppose that this orbifold has no essential homotopy annulus or equivalently it admits a complete hyperbolic structure. The end orbifolds have Euler characteristic equal to zero and all the singularities are of order p 3. Then we have Def E (O) = SDef E,ce (O). (2) and hol maps Def E (O) as an onto-map a component of representations rep E (π 1 (O), PGL(4, R)) which is also a component of rep i E,ce (π 1(O), PGL(4, R)).

90 43/46 Classification of the ends The end classification Let E be an end orbifold. Can the end be classified? We consider the case when E is not affine nor properly convex. Let E be a closed and convex but not properly convex real projective (n 1)-orbifold arising as an end of a IPC-orbifold of dimension n. The universal cover Ẽ of E is foliated by affine subspaces Rm with common boundary RP m 1. There is a projection p : Ẽ K where K is a properly convex open domain in RP n m 1.

91 Classification of the ends The end classification Let E be an end orbifold. Can the end be classified? We consider the case when E is not affine nor properly convex. Let E be a closed and convex but not properly convex real projective (n 1)-orbifold arising as an end of a IPC-orbifold of dimension n. The universal cover Ẽ of E is foliated by affine subspaces Rm with common boundary RP m 1. There is a projection p : Ẽ K where K is a properly convex open domain in RP n m 1. Let Γ be a subgroup of PGL(n, R) and the deck transformation group of Ẽ. Then there is an exact sequence 1 ker Γ Γ(K ) 1 where Γ(K ) acts on K divisibly or sweep K up. We can show that there is a map Γ Γ f PGL(m 1) by restricting the maps to RP m 1. Then we can show that the image Γ f has only eigenvalues of absolute value 1 only. Thus Γ f is virtually Solvable as shown by Fried mainly. (I think that this is true.) 43/46

92 Classification of the ends The end classification Let E be an end orbifold. Can the end be classified? We consider the case when E is not affine nor properly convex. Let E be a closed and convex but not properly convex real projective (n 1)-orbifold arising as an end of a IPC-orbifold of dimension n. The universal cover Ẽ of E is foliated by affine subspaces Rm with common boundary RP m 1. There is a projection p : Ẽ K where K is a properly convex open domain in RP n m 1. Let Γ be a subgroup of PGL(n, R) and the deck transformation group of Ẽ. Then there is an exact sequence 1 ker Γ Γ(K ) 1 where Γ(K ) acts on K divisibly or sweep K up. We can show that there is a map Γ Γ f PGL(m 1) by restricting the maps to RP m 1. Then we can show that the image Γ f has only eigenvalues of absolute value 1 only. Thus Γ f is virtually Solvable as shown by Fried mainly. (I think that this is true.) 43/46