arXiv research

New examples show some convex-cocompact subgroups are separable.

Characterizes convex cocompact actions in projective space with dynamical properties.

Convex-cocompact groups in infinite hyperbolic space are deformable.

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…

Characterizes Coxeter groups with convex cocompact representations in projective space.

Proves certain subgroups of genus 2 handlebody group are convex cocompact.

New subgroup behavior in genus-2 mapping class group identified.

Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.

Two groups with specific limit sets in hyperbolic spaces are identified.

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.

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…

Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.

Study on surface group representations in PU(2,1) leading to convex-cocompact examples.

Combination theorems for convex projective geometry subgroups.

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.

Constructs hyperbolic reflection groups with 3D limit sets.

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.

Study shows certain subgroups of fibered 3-manifolds are convex cocompact.

Frame flows on certain symmetric spaces mix exponentially.

In this paper we show that a given set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3-manifolds with that specified length spectrum, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.

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…

A Kleinian group $Γ< \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $Γ$ in $\mathbb H^3$ is quasiconvex or, equivalently, $Γ$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which…

New proof for certain groups in higher dimensions.

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…

Proves EGF representations in specific geometric contexts.

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 $x,y \in Teich(S)$ we obtain asymptotic formulas as $R\to \infty$ of $|B_{R}(x)\cap Gy|$ 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…

Sharp growth tightness proven for group quotients.

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 $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …

New spaces found without certain actions, using special subgroups.

Investigates properties of volume, entropy, and diameter in higher Teichmüller spaces.

Let $M$ be a convex cocompact acylindrical hyperbolic 3-manifold of infinite volume, and let $M^*$ denote the interior of the convex core of $M$. In this paper we show that any geodesic plane in $M^*$ is either closed or dense. We also show that only countably many planes are closed. These are the first rigidity theore…

New statistical convex-cocompactness found for non-orientable surfaces.

Maps between Hadamard manifolds are quasi-isometric to harmonic maps.

In this paper we prove that groups as in the title are convex cocompact in the mapping class group.

New Teichmüller spaces found for higher-dimensional groups.

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…

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_ρ$ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation $ρ$ of $π_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that…

The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduc…

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

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 $\text{Mod}(S)$. 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.

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 study constructs Yamabe operators on OC manifolds and proves their properties.

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…