Strongly Quasiconvex subgroups in 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.