2013/04/30 by Gilbert Levitt · 1 citation
Mathematics · #math.GR #math.GT
published as Ann. Inst. Fourier (Grenoble) 65 (2015), no. 2, 725--762 · Final version, to appear in Annales Institut Fourier
arxiv created 2014/05/15 · arxiv updated 2019/06/07
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 Fn. 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 F2), and we solve the commensurability problem (deciding whether two groups have isomorphic finite index subgroups) in a particular family of GBS groups.