Coarse geometry of extended admissible groups

TL;DR

This paper investigates the coarse geometric properties of extended admissible groups, showing their quasi-isometry invariance, nonpositive curvature under certain conditions, and analyzing their divergence and subgroup structures.

Contribution

It extends the understanding of coarse geometry for extended admissible groups, including invariance results, curvature properties, boundary behaviors, and divergence calculations.

Findings

Changing gluing isomorphisms does not affect quasi-isometry.

Extended admissible groups can exhibit large-scale nonpositive curvature.

The divergence of these groups is computed, generalizing previous results.

Abstract

Extended admissible groups belong to a particular class of graphs of groups that admit a decomposition generalizing those of non-geometric 3-manifold groups and Croke-Kleiner admissible groups. In this paper, we study several coarse-geometric aspects of extended admissible groups. We show that changing the gluing edge isomorphisms does not affect the quasi-isometry type of these groups. We also prove that, under mild conditions on the vertex groups, extended admissible groups exhibit large-scale nonpositive curvature, thereby answering a question posed by Nguyen-Yang. As an application, our results enlarge the class of extended admissible groups known to admit well-defined quasi-redirecting boundaries, a notion recently introduced by Qing-Rafi. In addition, we compute the divergence of extended admissible groups, generalizing a result of Gersten from non-geometric 3-manifold groups to this broader setting. Finally, we study several aspects of subgroup structure in extended admissible groups.