We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong negative answer to the question of whether being an HHG is invariant under quasi-isometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.
1
University of Bristol; University of Oxford, UK
2
Christ Church College, Saint Aldates; University of Oxford, UK
Harry Petyt; Davide Spriano. Unbounded domains in hierarchically hyperbolic groups. Groups, geometry, and dynamics, Tome 17 (2023) no. 2, pp. 479-500. doi: 10.4171/ggd/706
@article{10_4171_ggd_706,
author = {Harry Petyt and Davide Spriano},
title = {Unbounded domains in hierarchically hyperbolic groups},
journal = {Groups, geometry, and dynamics},
pages = {479--500},
year = {2023},
volume = {17},
number = {2},
doi = {10.4171/ggd/706},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/706/}
}
TY - JOUR
AU - Harry Petyt
AU - Davide Spriano
TI - Unbounded domains in hierarchically hyperbolic groups
JO - Groups, geometry, and dynamics
PY - 2023
SP - 479
EP - 500
VL - 17
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/706/
DO - 10.4171/ggd/706
ID - 10_4171_ggd_706
ER -
%0 Journal Article
%A Harry Petyt
%A Davide Spriano
%T Unbounded domains in hierarchically hyperbolic groups
%J Groups, geometry, and dynamics
%D 2023
%P 479-500
%V 17
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/706/
%R 10.4171/ggd/706
%F 10_4171_ggd_706
