Zariski Closures and Subgroup Separability

TL;DR

This paper refines subgroup separability results for free and surface groups by demonstrating finite dimensional representations that distinguish subgroups via Zariski topology, providing polynomial bounds on quotient sizes.

Contribution

It introduces a new approach using Zariski topology and finite dimensional representations to improve subgroup separability results for free and surface groups.

Findings

Finite dimensional representations separate subgroups in free and surface groups.

Polynomial upper bounds established for quotient sizes in subgroup separation.

Refinement of classical subgroup separability theorems.

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.