Study convex cocompact subgroups in real projective geometry.
problem Characterize discrete subgroups acting on real projective space.
method Define and characterize convex cocompactness, extend results from orthogonal groups.
result Equivalence of different convex cocompactness conditions for word 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.
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.
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.
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.
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.
New theorem about limit points in symmetric spaces.
problem Understanding limit points in symmetric spaces.
method Analyzing Zariski dense discrete subgroups and convex cocompact groups.
result Every limit point of a convex cocompact subgroup is conical.
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 …
Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.
problem Spectral gap for convex cocompact hyperbolic surfaces and their covers.
method Using thermodynamic formalism for twisted Selberg zeta functions.
result Uniform resonance-free regions for convex cocompact hyperbolic surfaces and expanders.
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.
Finite number of hyperbolic 3-manifolds for given lengths.
problem Counting hyperbolic 3-manifolds with specific geodesic lengths.
method Analyzing lengths of closed geodesics on convex cocompact manifolds.
result Only finitely many manifolds for specified lengths.
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.
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.
Study of Anosov representations in pseudo-Riemannian hyperbolic spaces.
problem Understanding Anosov representations in higher-dimensional spaces.
method Examining representations into projective indefinite orthogonal groups and their action on H^{p,q-1}.
result Intimate connection between Anosov representations and convex cocompactness in this setting.
A vanishing theorem for a convex cocompact hyperbolic manifold is established, which relates the L2 cohomology to the Hausdorff dimension of the limit set. The borderline case is shown to characterize the manifold completely.
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.
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.
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…
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…
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.
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.
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.
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.
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.
Study groups with contracting elements using SCC actions.
problem Understanding the asymptotic geometry of groups with contracting elements.
method Exploiting an extension lemma to prove properties of SCC actions.
result Groups with SCC actions contain large free sub-semigroups, have purely exponential growth, and have barrier-free sets with a growth-tight property.
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.
The paper proves properties of geodesic planes in a specific type of 3-manifold.
problem Characterizing geodesic planes in a specific class of 3-manifolds.
method Analyzing geodesic planes within the convex core of a hyperbolic 3-manifold.
result Geodesic planes are either closed or dense, with only countably many being closed.
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…
The paper shows how certain projective representations act on convex domains.
problem Understanding the action of projective Anosov representations on convex domains.
method Analyzing projective Anosov representations and their actions on properly convex domains in real projective space.
result Projective Anosov representations act convex cocompactly on properly convex domains.
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.
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 …
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)). Generalizes Nielsen equivalence theorem to hyperbolic group extensions.
problem Tackles Nielsen equivalence in hyperbolic group extensions.
method Generalizes a theorem by Juan Souto to a broader class of hyperbolic extensions.
result Includes all hyperbolic extensions of surfaces groups and free groups by Out$(F_n).
Study infinite subgroups of higher rank Lie groups, focusing on Anosov subgroups.
problem Understanding properties of Anosov subgroups in higher rank semisimple Lie groups.
method Characterize Anosov subgroups through geometric, coarse geometric, and dynamical viewpoints.
result New equivalent characterizations of Anosov subgroups, capturing rank one behavior.
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.
Classical-quantum correspondence on hyperbolic manifolds established.
problem Classical-quantum spectrum correspondence on hyperbolic manifolds.
method Direct correspondence between geodesic flow and Laplacian spectra.
result Established correspondence for convex cocompact hyperbolic manifolds.
The paper explores generic free subgroups and statistical hyperbolicity in group actions.
problem Understanding generic behavior of group actions and their implications.
method Analyzing statistically convex-cocompact actions and contracting elements.
result Exponential generic sets generate quasi-isometrically embedded free subgroups.
Proves stability pulls back under proper actions, with applications to mapping class groups and free groups.
problem Stability of subgroups in mapping class groups and free groups.
method Proves stability pulls back under proper actions on metric spaces.
result Stability of convex cocompact subgroups in mapping class groups and free groups.
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.
The paper proves a quantitative Tits alternative for negatively pinched manifolds.
problem Proving a quantitative version of the Tits alternative for negatively pinched manifolds.
method Analyzing discrete isometry subgroups generated by two non-elliptic isometries.
result A free subgroup of rank 2 is found in the isometry subgroup, which is convex-cocompact when one of the generators is hyperbolic.
In this paper we prove that groups as in the title are convex cocompact in the mapping class group.