Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
New examples show some convex-cocompact subgroups are separable.
problem Whether all convex-cocompact subgroups are separable.
method Using Manning-Mj-Sageev construction, examples of separable subgroups of arbitrary finite rank are given.
result Examples of separable convex-cocompact subgroups of arbitrary finite rank exist.
Characterizes convex cocompact actions in projective space with dynamical properties.
problem Understanding convex cocompact group actions in projective space.
method Dynamical characterization and expansion property analysis.
result Equivalence of convex cocompactness to an expansion property in different Grassmannians.
Anosov subgroups generalize convex-cocompact groups in hyperbolic geometry.
problem Understanding convex-cocompact subgroups in higher rank geometry.
method Characterizing Anosov subgroups and comparing them to convex-cocompact groups.
result Anosov subgroups are the right generalizations of convex-cocompact groups in hyperbolic geometry.
Characterizes Coxeter groups with convex cocompact representations in projective space.
problem Understanding representations of Coxeter groups as convex cocompact reflection groups.
method Investigates representations of Coxeter groups into GL(n,R) as geometric reflection groups in projective space.
result Characterizes Coxeter groups that admit convex cocompact representations and describes the spaces of such representations.
We study a notion of convex cocompactness for discrete subgroups of the projective general linear group acting (not necessarily irreducibly) on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good pr…
Proves certain subgroups of genus 2 handlebody group are convex cocompact.
problem Characterizing subgroups of genus 2 handlebody group.
method Proving convex cocompactness of purely pseudo-Anosov subgroups.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact.
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.
problem Understanding the structure of pseudo-Anosov subgroups in surface bundles over tori.
method Using the Birman exact sequence to show convex cocompactness.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact in surface bundles over tori.
Two groups with specific limit sets in hyperbolic spaces are identified.
problem Identifying convex cocompact subgroups with specific limit sets in real hyperbolic spaces.
method Examples of subgroups generated by reflections and rotations with limit sets as Pontryagin spheres and Menger curves.
result Examples of convex cocompact subgroups with limit sets as Pontryagin spheres and Menger curves are found.
We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. That is, if the right-angled Artin group G in Mod(S) satisfies certain conditions that imply G is quasi-isometrically embedded in Mod(S), then a purely pseudo-Anosov subgroup H of …
Constructs hyperbolic reflection groups with 3D limit sets.
problem Existence of convex cocompact groups with specific limit sets.
method Inputting a simplicial complex into a construction process yields a hyperbolic reflection group.
result Answers Kapovich's question affirmatively by creating a thin subgroup of an arithmetic lattice.
Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.
problem Understanding Anosov representations and their properties.
method Characterizations via equivariant limit maps, Cartan property, and uniform gap summation.
result Characterizations of Anosov representations and strongly convex cocompact subgroups.
Combination theorems for convex projective geometry subgroups.
problem Understanding discrete subgroups in convex projective geometry.
method General combination theorems for discrete subgroups preserving properly convex open subsets.
result Free products of convex cocompact subgroups are convex cocompact.
Study shows certain subgroups of fibered 3-manifolds are convex cocompact.
problem Understanding subgroups of fibered 3-manifolds in mapping class groups.
method Used the Birman exact sequence to show convex cocompactness.
result Finitely generated pseudo-Anosov subgroups are convex cocompact.
A Kleinian group Γ<Isom(H3) is called convex cocompact if any orbit of Γ in H3 is quasiconvex or, equivalently, Γ acts cocompactly on the convex hull of its limit set in ∂H3. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which…
Study on surface group representations in PU(2,1) leading to convex-cocompact examples.
problem Nonmaximal representations of surface groups in PU(2,1).
method Analysis of convex-cocompact representations with unique equivariant minimal surfaces.
result Existence of convex-cocompact representations with specific properties.
We show that every limit point of a Zariski dense discrete subgroup Γ of the isometry group of a symmetric space of noncompact type is conical if and only if Γ is convex cocompact.
We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.
We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic group then G i…
Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not a…
New proof for certain groups in higher dimensions.
problem Properties of discrete subgroups in higher dimensions.
method Proving convex-cocompactness for specific groups.
result Finitely generated Kleinian groups with small critical exponent are convex-cocompact.
We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompac…
Sharp growth tightness proven for group quotients.
problem Growth behavior of group quotients by confined subgroups.
method Statistically convex-cocompact action with contracting elements.
result Sharp growth tightness proven, with applications to uniformly recurrent subgroups.
New statistical convex-cocompactness found for non-orientable surfaces.
problem Understanding the dynamics of mapping class groups on non-orientable surfaces.
method Using Teichmüller space and complexity length, showing geodesics leave compact regions with exponentially low probabilities.
result Statistical convex-cocompactness of mapping class groups on non-orientable surfaces.
Proves EGF representations in specific geometric contexts.
problem Understanding representations of groups with hyperbolic properties.
method Analyzes projectively convex cocompact manifolds and convex projective manifolds with generalized cusps.
result Holonomy representations of specific geometric manifolds are EGF representations.
We strengthen the analogy between convex co-compact Kleinian groups and convex co-compact subgroups of the mapping class group of a surface (in the sense of B. Farb and L. Mosher).
Complex hyperbolic Kleinian groups yield Stein manifolds under certain conditions.
problem Characterizing discrete groups acting on complex hyperbolic spaces.
method Proving conditions for a discrete group to yield a Stein manifold.
result If a discrete group is convex-cocompact, torsion-free, and has a critical exponent less than 2, the quotient manifold is Stein.
We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of Out(Fn) as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …
In this paper we prove that groups as in the title are convex cocompact in the mapping class group.
Investigates properties of volume, entropy, and diameter in higher Teichmüller spaces.
problem Properties of volume, entropy, and diameter for representations in mSO(p,q+1). method Uniform lower bound on entropy times volume, upper bound on entropy, finiteness and compactness results.
result Entropy is bounded by p−1 for representations conjugate to mS(mO(p,1)imesmO(q)). This paper presents a study of the asymptotic geometry of groups with contracting elements, with emphasis on a subclass of statistically convex-cocompact (SCC) actions. The class of SCC actions includes relatively hyperbolic groups, CAT(0) groups with rank-1 elements and mapping class groups, among others. We exploit a…
It is known that every infinite index quasi-convex subgroup H of a non-elementary hyperbolic group G is a free factor in a larger quasi-convex subgroup of G. We give a probabilistic generalization of this result. That is, we show that when R is a subgroup generated by independent random walks in G, then $\lan…
The paper classifies Kleinian groups with Hausdorff dimension less than 1.
problem Classifying Kleinian groups with specific Hausdorff dimensions.
method Using Hou's result, the paper proves that all convex cocompact Kleinian groups of Hausdorff dimension less than 1 are Schottky groups.
result The classification of convex cocompact Kleinian groups of Hausdorff dimensions less than 1.
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
New spaces found without certain actions, using special subgroups.
problem Existence of proper actions on homogeneous spaces.
method Using convex cocompact representations and nilpotent orbits theory.
result Found new homogeneous spaces without specific actions.
For a convex cocompact subgroup G<Mod(S), and points x,y∈Teich(S) we obtain asymptotic formulas as R→∞ of ∣BR(x)∩Gy∣ as well as the number of conjugacy classes of pseudo-Anosov elements in G of dilatation at most R. We do this by developing an analogue of Patterson-Sullivan theory for the…
We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder cond…
Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.
problem Understanding continuous paths in discrete subgroups of hyperbolic space.
method Combination theorem and chromatography technique.
result Construction of an exotic path of discrete subgroups with no isomorphic subgroups.
Study contractibility of boundaries in convex sets and limit sets of subgroups.
problem Understanding contractibility of boundaries and wildness of limit sets in geometric structures.
method Use sufficient conditions for contractibility, study coarse upper curvature bounds, and analyze interpolation in geodesic metric spaces.
result Conditions for contractibility of boundaries and properties of limit sets are established.
New Teichmüller spaces found for higher-dimensional groups.
problem Finding new Teichmüller spaces for higher-dimensional groups.
method Proving representations of groups in pseudo-Riemannian hyperbolic spaces are convex cocompact.
result Set of representations forms connected components of Hom spaces.
The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface S is a function Eρ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation ρ of π1(S). Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that…
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …
We prove Patterson's conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.
New representations for surface groups in PU(2,1) are stable and larger than convex cocompact ones.
problem Characterizing representations of surface groups in PU(2,1).
method Introducing simple-stable representations and proving their properties.
result The set of conjugacy classes of simple-stable representations is a domain of discontinuity strictly larger than convex cocompact representations.
There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answe…
The paper proves a unique conformal measure for Anosov groups and shows local mixing.
problem Proving the uniqueness of conformal measures for Anosov groups.
method Analogue of Sullivan's theorem for Anosov subgroups of semisimple groups.
result Uniqueness of conformal measures and local mixing for Anosov groups.