Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness

Abstract

We prove that a group acting geometrically on a product of proper minimal CATp´1q spaces without permuting isometric factors is a hierarchically hyperbolic group. As an application we construct hierarchically hyperbolic groups which are not virtually torsion-free.

Mapping class groups are quasicubical

“…We say that an HHG pG, Sq is colourable if there is a finite partition S " Ů χ i"1 S i such that G acts by permutations on tS i : 1 ď i ď χu and any two elements of any one S i are transverse. Note that G need not be virtually torsionfree [Hug21]. The important part about the colouring for us is that it gives access to the following.…”

Irreducible lattices fibring over the circle

“…Also note that a number of group presentations and results regarding residual finiteness, 2 -Betti numbers, and autostackability will only exist in the thesis version. Finally, we remark the existence of the other companion paper [Hug21c] where the author constructs a lattice (and in fact a hierarchically hyperbolic group) in a product of trees which is not virtually torsion-free.…”