# Alternating quotients of right-angled Coxeter groups

## TL;DR

This paper characterizes when right-angled Coxeter groups have certain alternating quotients, linking graph connectivity to subgroup separability, and applies these ideas to right-angled Artin groups and hyperbolic surface groups.

## Contribution

It establishes a new criterion connecting graph connectivity with the existence of alternating quotients for right-angled Coxeter groups and related groups.

## Key findings

- Connectedness of the complement graph characterizes alternating quotients.
- Right-angled Artin groups can be decomposed into factors with many alternating quotients.
- Finitely generated subgroups of hyperbolic surface groups can be separated from finite sets in alternating quotients.

## Abstract

Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group such that $f (\gamma_i) \notin f (H)$. A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense. Similarly, finitely generated subgroups of closed, orientable, hyperbolic surface groups can be separated from finitely many elements in an alternating quotient, answering positively a conjecture of Wilton.