Virtual retractions, conjugacy separability and omnipotence

TL;DR

This paper develops criteria using wreath products for groups to be conjugacy separable or omnipotent, applying these to prove properties of Fuchsian and surface groups.

Contribution

It introduces new criteria based on virtual retractions onto cyclic subgroups and applies them to key classes of groups.

Findings

Infinite-order elements of certain Fuchsian groups are conjugacy distinguished.

Surface groups are proven to be omnipotent.

Provides a topological proof of Stebe's theorem.

Abstract

We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.