Vertex finiteness for splittings of relatively hyperbolic groups

TL;DR

This paper proves vertex finiteness for splittings of certain relatively hyperbolic groups over specific families of subgroups, showing only finitely many vertex stabilizers up to isomorphism in these cases.

Contribution

It establishes vertex finiteness results for splittings of toral relatively hyperbolic groups and groups hyperbolic relative to virtually polycyclic subgroups, extending understanding of their structural decompositions.

Findings

Vertex finiteness holds for toral relatively hyperbolic groups over abelian subgroups.

Vertex finiteness holds for groups hyperbolic relative to virtually cyclic subgroups.

Finitely many minimal G-trees with virtually cyclic edge stabilizers for one-ended groups.

Abstract

Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G is a toral relatively hyperbolic group and $\mathcal{A}$ is the family of abelian subgroups. We also show vertex finiteness when G is hyperbolic relative to virtually polycyclic subgroups and $\mathcal{A}$ is the family of virtually cyclic subgroups; if moreover G is one-ended, there are only finitely many minimal G-trees with virtually cyclic edge stabilizers, up to automorphisms of G.