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.
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.
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 characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
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…
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.
Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.
problem Investigate non-standard bi-orders on punctured torus bundles.
method Analyze various bi-orderings and compare them to standard ones formed by the lower central series.
result For every bi-ordering, the largest and second largest proper convex subgroups match those of a standard bi-ordering. Third largest subgroup matches if it exists.
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.
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 …
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.
The paper provides conditions for amalgamation of certain subgroups and preserves convexity properties.
problem Conditions for amalgamation of subgroups in hierarchically hyperbolic groups.
method Study of amalgamation conditions and preservation of convexity properties.
result Conditions under which amalgamation preserves hierarchical quasiconvexity and strong quasiconvexity.
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.
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.
Study subgroups preserving proper domains in flag manifolds.
problem Identify subgroups preserving proper domains in flag manifolds.
method Establish necessary conditions and introduce causal convexity in Shilov boundary.
result Transverse subgroups with geometric properties in Shilov boundary.
Counting subgroups of a surface using convex core lengths.
problem Counting conjugacy classes of subgroups of fundamental groups of surfaces.
method Using half the sum of the lengths of the boundaries of the convex core of a subgroup.
result The number of conjugacy classes of subgroups is asymptotic to cL6g−6+2r. We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of ran…
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 of groups and their quasi-isometrically embedded subgroups.
problem Understanding the structure and properties of groups and their subgroups.
method Abstracting the notion of A/QI triples and using methods from geometric group theory.
result Stability of quasi-isometrically embedded subgroups in finitely generated groups.
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 …
Geometric limits of cyclic subgroups in specific groups studied.
problem Understanding geometric limits of cyclic subgroups in SO_0(1, k+1) and SU(1, k+1).
method Construction of sequences of subgroups and analysis of their geometric limits.
result Examples of sequences with geometric limits strictly containing algebraic limits.
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.
Study surface subgroups acting on projective space, finding bending laminations and spheres.
problem Surface subgroups acting on RP3 with coaffine representations. method Stratification of convex core boundary, bending laminations, and analysis of holonomy.
result Projectivization of bending data space is a sphere of dimension 6g−7. The convex-cocompact subgroups are central in hyperbolic geometry and more generally in negative curvature. Labourie introduced in 2005 the notion of 'Anosov' subgroup which proves progressively to be the right generalizations of convex-cocompact groups, especially after the works of Kapovich, Leeb and Porti. This expo…
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
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.
Study growth rates of subgroups in groups with a constricting element.
problem Understanding growth rates of subgroups in groups with a constricting element.
method Examining the spectrum of relative and quotient exponential growth rates of quasi-convex subgroups.
result Determine when growth rates of subgroups are strictly smaller or coincide with the group's growth rate.
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 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.
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.
The study characterizes subgroups of mapping tori of free groups.
problem Characterizing subgroups of mapping tori of free groups.
method Using canonical maximal sub-mapping tori and relative hyperbolicity.
result Characterizes locally quasi-convex hyperbolic groups.
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.
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…
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).
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…
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 show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group G is Morse, that is to say any quasi-geodesic connecting points in the cyclic group C generated by g stays close to C. This answers a question of Dahmani-Guirardel-Osin. What is more, we show that hyper…
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.
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.
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, t…
The study extends Dehn filling to Lie groups, ensuring geometric properties.
problem Generalizing Dehn filling to semisimple Lie groups.
method Analyzing deformations of subgroups and their geometric properties.
result Extended geometrically finite subgroups can be deformed while maintaining properties.
The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of …
Combination theorem for geodesic coarsely convex group pairs.
problem Understanding properties of groups relative to subgroups.
method Definitions of weakly semihyperbolic, semihyperbolic, and geodesic coarsely convex group pairs; combination theorem.
result Combination theorem for geodesic coarsely convex group pairs.
For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class …
In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds X. Precisely, we prove that a nonelementary discrete isometry subgroup of Isom(X) generated by two non-elliptic isometries g, f contains a free subgroup of rank 2 generated by isometries fN,h …
Anosov subgroups' deformations affect limit cones and growth indicators continuously.
problem Understanding continuous changes in Anosov subgroups' effects on limit cones and growth indicators.
method Continuous variation of limit cones and growth indicators under deformations of Anosov subgroups, with convexity assumptions.
result Limit cones and growth indicators vary continuously under deformations of Anosov subgroups.
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.
Affine deformations of convex cones on projective surfaces.
problem Understanding affine actions on convex cones.
method Geometric correspondence and convex tube domains.
result Quotients of convex domains are affine manifolds with convex surfaces.