Zariski closures and subgroup separability

Abstract

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate a finitely generated subgroup in a free or surface group. Mathematics Subject Classification 20E05 · 20E26

Effective subgroup separability of finitely generated nilpotent groups

“…Free groups and surface groups. -Let G be a free group of rank r > 1 or a surface group of genus g > 1 containing a finitely generated subgroup H. In [35], Louder, McReynolds, and Patel construct a representation ρ H : G → GL(V ) such that the subgroup ρ H (H) is closed for the subspace topology on ρ(G) induced by the Zariski topology on GL(V ). As a consequence, they find the following result on effective separability.…”