Finite non-parabolic subgroups of relatively hyperbolic groups

Abstract: Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups ${H_{\lambda}}{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We give an upper bound on the orders of finite non-parabolic subgroups of $G$ in terms of some fundamental constants associated with $\mathcal{P}$. This upper bound is computable if $G$ is finitely generated and the word problem in each $H{\lambda}$, $\lambda\in \Lambda$, is decidable.