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.
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.
Quantum resonances for tensors on hyperbolic spaces are studied.
problem Quantum resonances of symmetric tensors on asymptotically hyperbolic spaces.
method Analyzes the Lichnerowicz Laplacian on manifolds with even Riemannian conformally compact Einstein metrics and quotients of hyperbolic space.
result Resolvent of the Lichnerowicz Laplacian has meromorphic continuation to the complex plane, defining quantum resonances.
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.
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 establishes a new lower bound for limit set dimensions.
problem Understanding the Hausdorff dimension of limit sets in symmetric spaces.
method Analyzing tangents of Lipschitz differentiability spaces embedded in Carnot groups.
result Equality in the lower bound is achieved precisely when the group stabilizes a copy of a symmetric space.
The bending map of a hyperbolic 3-manifold maps a convex cocompact hyperbolic metric on a hyperbolic 3-manifold with boundary to its bending measured geodesic lamination. In the present paper we study the extension of this map to the space of geometrically finite hyperbolic metrics. We introduce a relationship on the s…
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…
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.
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.
A Kleinian manifold Y is a quotient of a rank-one symmetric space of non-compact type by a convex-cocompact discrete group of isometries. We describe the spectral decomposition of the space of square integrable sections of locally homogeneous bundles on Y with respect to locally invariant differential operators. In the…
Let Y=Γ\Hn be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of Γ-invariant currents on the sphere at infinity of Hn with support on the limit set of Γ. These spaces are finite-dimensional. The main result identifies th…
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.
We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this …
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.
Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex coc…
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.
Holomorphic curves found in compact quotients of SL(2,C).
problem Existence of holomorphic curves in compact quotients of SL(2,C).
method Proved the existence of a pair (Σ, Γ) with specific properties.
result Affirmative answer to questions raised by Huckleberry and Winkelmann, Ghys.
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.
The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replace…
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.
We obtain the Plancherel theorem for the quotient of a simple Lie group of real rank one by a convex-cocompact discrete subgroup and its consequences for the spectrum of locally invariant differential operators on bundles over Kleinian manifolds. We develop a geometric version of scattering theory. The paper is an upda…
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.
Generalizes entropy-drift inequality for specific geometric spaces.
problem Entropy, drift, and critical exponent in Gibbs measures on geometrically finite manifolds.
method Generalization of Guivarc'h's inequality for CAT(-1) spaces, analysis of random walks.
result Equality in entropy-drift inequality achieved if and only if Gibbs density is equivalent to hitting measure.
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.
Paper finds new ball quotients from curve products.
problem Finding rational deformations of surfaces.
method Using cocompact lattices in PU(2,1).
result Shows existence of new ball quotients for surfaces.
Researchers extend parametrization of Margulis spacetimes using strip deformations.
problem Parametrize Margulis spacetimes with decorated horoballs.
method Use strip deformations to parametrize complete finite-area hyperbolic surfaces with spikes decorated with horoballs.
result Generalized parametrization of Margulis spacetimes with photons.
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.
Finite groups generate only finitely many Kleinian groups.
problem Characterizing finite groups generating Kleinian groups.
method Generalized Gromov--Guth inequality and orbifold volume bounds.
result Only finitely many conjugacy classes of two-generated arithmetic Kleinian groups.
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.
Let H be a discrete cocompact subgroup of SL_2(C). We conjecture that the quotient manifold X=SL_2(C)/H contains infinitely many non-isogeneous elliptic curves and prove that this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where the intersection of H and SL_2(R) is cocompact…