Let be a right-angled Coxeter group corresponding to a finite non-discrete graph with at least vertices. Our main theorem says that is connected if and only if for any infinite index convex-cocompact subgroup of and any finite subset there is a surjection from to a finite alternating group such that . 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 the conjecture of Wilton .
Cite this article
Michal Buran, Alternating quotients of right-angled Coxeter groups. Groups Geom. Dyn. 15 (2021), no. 3, pp. 965–987DOI 10.4171/GGD/617