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Conjugator lengths in hierarchically hyperbolic groups
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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.
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Cited by 1 Pith paper
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Hierarchically hyperbolic groups and uniform exponential growth
A virtually torsion-free hierarchically hyperbolic group either has uniform exponential growth or its Cayley graph is quasi-isometric to Z times a space.
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