Hierarchical hyperbolicity of hyperbolic-2-decomposable groups

Abstract

Let G be a graph of hyperbolic groups with 2-ended edge groups. We show that G is hierarchically hyperbolic if and only if G has no distorted infinite cyclic subgroup. More precisely, we show that G is hierarchically hyperbolic if and only if G does not contain certain quotients of Baumslag-Solitar groups. As a consequence, we obtain several new results about this class, such as quadratic isoperimetric inequality and finite asymptotic dimension.

Extensions of multicurve stabilizers are hierarchically hyperbolic

“…Hierarchical hyperbolicity axiomatizes this theory, describing a class of spaces whose coarse geometry is encoded in a collection of projections onto hyperbolic metric spaces that are organized by a set of combinatorial relations. Remarkably, the class of hierarchically hyperbolic spaces encompasses a variety of groups beyond the mapping class group including the fundamental group of most 3-manifolds [BHS19], many cocompactly cubulated groups [BHS17b,HS20], Artin groups of extra large type [HMS], and several combinations of hyperbolic groups [BR20a,RS,BR20b]. Hierarchical hyperbolicity also describes the coarse geometry of a number of other groups and spaces associated to surfaces such as Teichmüller space with both the Teichmüller and Weil-Peterson metrics [BHS17b, MM99, MM00, BKMM12, Bro03, Dur16, Raf07, EMR17], the genus 2 handlebody group [Che20], the π 1 pSq-extensions of lattice Veech groups [DDLS], certain quotients of the mapping class group [BHS17a,BHMS20], and a wide variety of graphs built from curves on surfaces [Vok17].…”