A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups

TL;DR

This paper establishes conditions under which subgroups of hierarchically hyperbolic groups form amalgamated free products, with applications to geometric subgroups of mapping class groups, and investigates the preservation of convexity properties during amalgamation.

Contribution

It provides new sufficient conditions for subgroup amalgamation in hierarchically hyperbolic groups and analyzes the preservation of convexity notions in this context.

Findings

Amalgamation conditions for hierarchically hyperbolic groups.

Application to geometric subgroups of mapping class groups.

Preservation of hierarchical quasiconvexity and strong quasiconvexity during amalgamation.

Abstract

We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups of finite-type surfaces, that is, those subgroups coming from the embeddings of closed subsurfaces. In the second half of the paper, we study under which hypotheses our amalgamation procedure preserves several notions of convexity, such as hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in a uniform neighbourhood). This answers a question of Russell, Spriano, and Tran.