Strongly Quasiconvex subgroups in graphs of groups

TL;DR

This paper explores the properties of finitely generated subgroups in graphs of groups, establishing equivalences between finite height, strong quasiconvexity, and virtual freeness, and examines conditions for preserving strong quasiconvexity under amalgamation.

Contribution

It proves the equivalence of several subgroup properties in graphs of groups and provides criteria for the preservation of strong quasiconvexity during amalgamation.

Findings

Equivalence of finite height, strong quasiconvexity, and virtual freeness for subgroups.

Conditions for strong quasiconvexity preservation under amalgams.

Characterization of $A/QI$--triples in the context of graphs of groups.

Abstract

Given a graph of groups $\mathcal{G} = (\Gamma, \{G_v\}, \{G_e\})$ with certain conditions on vertex groups and $G$ acts acylindrically on its Bass-Serre tree $T$. Let $H$ be a finitely generated subgroup of $G$. We prove the following statements equivalence: $H$ has finite height, $(G, T, H)$ is a $A/QI$--triple, $H$ is strongly quasiconvex and virtually free in $G$. We also give a condition to determine whether strong quasiconvexity in a group is preserved under amalgams.