Wenyuan Yang

Growth tightness for groups with contracting elements

We establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

Peripheral structures of relatively hyperbolic groups

In this paper, we introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained using dynamical methods. Some applications are discussed. The class of groups acting geometrically finitely on Floyd boundaries turns out to be easily understood. However, we also show that Dunwoodys inaccessible group does not act geometrically finitely on its Floyd boundary.

Limit sets of relatively hyperbolic groups

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundaries.

Separable subgroups have bounded packing

In this note, we prove that separable subgroups have bounded packing in ambient groups. The notion bounded packing was introduced by Hruska and Wise and in particular, our result answers positively a question of theirs, asking whether each subgroup of a virtually polycyclic group has the bounded packing property.

LIMIT SETS AND COMMENSURABILITY OF KLEINIAN GROUPS

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups G 1 and G 2 of an infinite co-volume Kleinian group G ⊂Isom( H 3 ) having Λ( G 1 )=Λ( G 2 ) are commensurable. In particular, we prove that any finitely generated subgroup H of a Kleinian group G ⊂Isom( H 3 ) with Λ( H )=Λ( G ) is of finite index if and only if H is not a virtually fibered subgroup.

Genericity of contracting elements in groups

We establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. We obtain as corollaries results on the exponential genericity for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 elements in CAT(0) groups, and the set of pseudo-Anosov elements in mapping class groups. For a proper action with purely exponential growth, we show that the set of contracting elements is generic. In particular, for mapping class groups, the set of pseudo-Anosov elements is generic in a sufficiently large subgroup, provided that the subgroup has purely exponential growth. By Roblin’s work, we obtain that the set of hyperbolic elements is generic in any discrete group action on CAT(

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

-1

Growth tightness of groups with nontrivial Floyd boundary

- 1 ) space with finite BMS measure. We present applications to the number of conjugacy classes of non-rank-1 elements in CAT(0) groups with rank-1 elements.