Virtual specialness of the double

Abstract

Let $G$ be a virtually compact special Gromov-hyperbolic group. We prove that the double $G *_H G$ along a quasiconvex subgroup $H$ is virtually compact special. More generally, we show that if a finite graph of groups has constant vertex groups, with each vertex group virtually compact special Gromov-hyperbolic and each edge group quasiconvex in its adjacent vertex groups, then its fundamental group is virtually compact special.