Large profinite genus and extensions in Bestvina-Brady groups

Abstract

We construct the first examples of uncountable families of residually finite duality groups of type $FP$, all of which have isomorphic profinite completions, while being pairwise not quasi-isometric. To establish these finiteness properties, we prove conjectures of Leary regarding groups of deck transformations of certain cubical branched covers being residually finite and virtually special. The main new tool is a strong separability result for Bestvina-Brady groups, allowing the extension of any finite quotient of many Bestvina-Brady subgroups to a finite quotient of the whole group. This gives a finite version of the congruence extension property, which we also establish for the same group pairs.