Asymptotically rigid mapping class groups III: Presentations and isomorphisms
Anthony Genevois, Anne Lonjou, Christian Urech
TL;DR
The paper provides an explicit finite presentation for all braided Higman-Thompson groups $brT_{n,m}$ via Brown's method acting on a bounded-height spine cube complex $ extsc{SC}_{ullet}(A_{n,m})$, and it computes their abelianisations as $brT_{n,m}^{ab} \ Z_m \times Z_{|m-n+1|}$. It shows that the subgroup $B_ _ ty$ is characteristic and that any isomorphism $brT_{n,m} \to brT_{r,s}$ induces an isomorphism $T_{n,m} \to T_{r,s}$, enabling a Rubin-based constraint that forces $n=r$ and restricts $s$ to either $m$ or $|m-n+1|$ under certain conditions. The main contributions are the complete presentation, the abelianisation computation, and partial rigidity results for the isomorphism problem among these groups, advancing understanding of how asymptotically rigid mapping class groups differentiate braided Higman-Thompson variants. The work provides concrete algebraic invariants to distinguish braided Higman-Thompson groups and connects group-theoretic structure to the underlying topological action on the spine complex, with implications for the broader isomorphism problem in asymptotically rigid MCGs.
Abstract
This article is dedicated to the computation of an explicit presentation of some asymptotically rigid mapping class groups, namely the braided Higman-Thompson groups. To do so, we use the action of these groups on the spine complex, a simply connected cube complex constructed by the authors in a previous work. In particular, this allows to compute the abelianisations of these groups. With these new algebraic invariants we can handle many new cases of the isomorphism problem for asymptotically rigid mapping class groups of trees.