Conjugator lengths in hierarchically hyperbolic 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.