Generalized Baumslag-Solitar groups: rank and finite index subgroups

TL;DR

This paper develops methods to compute the rank of generalized Baumslag-Solitar groups, analyzes their finite index subgroups, and addresses the isomorphism problem within certain families, advancing understanding of their algebraic structure.

Contribution

It provides an effective way to determine the rank of GBS groups, analyzes their finite index subgroups, and solves the isomorphism problem for specific GBS group families.

Findings

Effective rank computation for GBS groups

Finite index subgroup rank bounds

Decidability of isomorphism in certain GBS groups

Abstract

A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite order outer automorphism of a free group $F_n$. We also show that the rank of a finite index subgroup of a GBS group G cannot be smaller than the rank of G. We determine which GBS groups are large (some finite index subgroup maps onto $F_2$), and we solve the commensurability problem (deciding whether two groups have isomorphic finite index subgroups) in a particular family of GBS groups.