The criterion for relative quasiconvexity of tamely generated subgroups

Let $G$ be hyperbolic relative to a peripheral structure $\mathbb P$. Suppose that $G$ splits as a finite graph of groups whose edge groups are finitely generated and relatively quasiconvex. Let $H\leq G$ be tamely generated, and suppose that each subgroup $H_v$ is finitely generated for every vertex $v$ in the Bass–Serre tree.

Relative quasiconvexity criterion. Then $H$ is relatively quasiconvex in $G$.

This criterion is suggested by the strongest result proved in the paper and would extend the established quasiconvexity results for subgroups arising from splittings. The statement is presented as a conjectural criterion; its general validity is not established here.