Conjugator lengths in hierarchically hyperbolic groups

TL;DR

This paper provides upper bounds on the length of shortest conjugators in hierarchically hyperbolic groups, including mapping class groups and right-angled Artin groups, with linear bounds for Morse elements and their powers.

Contribution

It establishes the first general linear bounds on conjugator lengths in hierarchically hyperbolic groups, extending to a broad class of important groups.

Findings

Linear bounds on conjugator lengths for Morse elements.

Sharper linear bounds for powers of conjugate elements in certain groups.

Applicable to groups like mapping class groups, right-angled Artin groups, and 3-manifold groups.

Abstract

In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger--Mozes-type groups, most $3$--manifold groups, and many others. In this setting we establish a linear bound on the length of the shortest conjugator for any pair of conjugate Morse elements. For a subclass of these groups, including, in particular, all virtually compact special groups, we prove a sharper result by obtaining a linear bound on the length of the shortest conjugator between a suitable power of any pair of conjugate infinite order elements.