On the automorphism group of generalized Baumslag-Solitar groups

Abstract

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If $G$ is unimodular (virtually $F_n\times Z$), then Out(G) is commensurable with a semi-direct product $Z^k⋊Out(H)$ with $H$ virtually free.

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Cited by in corpus (34)

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• Twisted Conjugacy Classes in Lattices in Semisimple Lie Groups
• Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups
• Invariants of orbit equivalence relations and Baumslag-Solitar groups
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• The Bass-Jiang group for automorphism-induced HNN-extensions
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• Non proper HNN extensions and uniform uniform exponential growth
• Model geometries of finitely generated groups
• Characteristic subgroups and the R${}_{\infty }$-property for virtual braid groups
• The R${}_{\infty }$ property for pure Artin braid groups
• The R${}_{\infty }$-property for braid groups over orientable surfaces
• On knot groups acting on trees
• Whitehead algorithm for automorphisms of generalized Baumslag-Solitar groups
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