On Properties of Relatively Hyperbolic Groups

Abstract

We discuss a number of problems in relatively hyperbolic groups. We show that the word problem and the conjugacy (search) problem are solvable in linear and quadratic time, respectively, for a relatively hyperbolic group, whenever the corresponding problem is solvable in linear and quadratic time in each parabolic subgroup. We also consider the class R of finitely generated toral relatively hyperbolic groups. We show that groups from R are commutative transitive and generalize a theorem proved by Baumslag in [Bau67] to R. Moreover, we discuss two definitions of (fully) residually-C groups, the classical Definition 1.2.1 and a modified Definition 1.2.5. Building upon results obtained by Ol'shanskii [Ol'93] and Osin [Osi10], we prove the equivalence of the two definitions for C = R. This is a generalization of the similar result obtained by Ol'shanskii for the class of torsion-free hyperbolic groups. Let Γ ∈ R be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [KM12] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from R. On the other hand, we give an example of a finitely generated torsion-free fully residually-H group that does not embed into a group from R; H is the class of hyperbolic groups.

Relatively Anosov representations via flows II: Examples

“…Remark Theorem 1.3 leaves open the question if the above conditions are equivalent to $$\rho$$ is being $$\operatorname{\mathsf {P}}_k$$‐Anosov relative to any weak cusp space. Using a different flow space (which is equivalent to ours when $$X$$ is $${\rm CAT}(-1)$$), Wang showed that this is the case [40]. …”