Conjugator lengths in hierarchically hyperbolic groups

In this paper we establish upper bounds on the length of the shortest conjugator between pairs of 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. One case of our result in this setting is a linear bound on the length of the shortest conjugator for any pair of conjugate Morse elements. In a special case, namely, for virtually compact special cubical groups, we can prove a sharper result by obtaining a linear bound on the length of the shortest conjugator between any pair of infinite order elements. In a more general case, that of acylindrically hyperbolic groups, we establish an upper bound on the length of shortest conjugators, but in this generality the bound may not be linear.