On subgroup conjugacy separability in the class of virtually free groups

TL;DR

This paper investigates subgroup conjugacy separability in certain groups, proving that free groups and some fundamental groups possess this property and introducing a related new property.

Contribution

It establishes subgroup conjugacy separability for free groups and certain fundamental groups, and introduces the subgroup into-conjugacy separability property.

Findings

Free groups are subgroup conjugacy separable.

Fundamental groups of finite trees of finite groups are subgroup conjugacy separable.

These groups also have the subgroup into-conjugacy separability property.

Abstract

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental groups of finite trees of finite groups with some normalizer condition are SCS. We also introduce the subgroup into-conjugacy separability property and prove that the above groups have this property too.