Vertex finiteness for splittings of relatively hyperbolic 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.

Minor modifications following referee's comments. To appear in Israel Journal of Mathematics

References in corpus (1)

• McCool groups of toral relatively hyperbolic groups

Cited by in corpus (2)

• JSJ decompositions of groups
• McCool groups of toral relatively hyperbolic groups