The Geometry of the Conjugacy Problem in Wreath Products and Free Solvable Groups

TL;DR

This paper investigates the conjugacy problem in wreath products and free solvable groups, providing bounds on conjugator lengths and analyzing how group properties influence conjugacy complexity.

Contribution

It introduces an effective approach to the conjugacy problem, establishing cubic bounds for free solvable groups and analyzing factors affecting wreath product conjugacy complexity.

Findings

Conjugacy length function for free solvable groups is at most cubic.

Behavior of conjugacy problem in wreath products depends on component groups.

Subgroup distortion impacts conjugacy length in wreath products.

Abstract

We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to the definition of the conjugacy length function. We show that for free solvable groups the conjugacy length function is at most cubic. For wreath products the behaviour depends on the conjugacy length function of the two groups involved, as well as subgroup distortion within the quotient group.