Virtual specialness of the double

TL;DR

The paper proves that certain complex group constructions, specifically doubles and graphs of groups with specific properties, are virtually compact special, extending the class of known such groups.

Contribution

It establishes that doubles of virtually compact special Gromov-hyperbolic groups along quasiconvex subgroups are virtually compact special, generalizing to graphs of groups with similar properties.

Findings

Doubles of the specified groups are virtually compact special.

Fundamental groups of certain graphs of groups are virtually compact special.

The results extend the class of known virtually compact special groups.

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.