#### Volume 11, issue 1 (2007)

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On the automorphism group of generalized Baumslag–Solitar groups

### Gilbert Levitt

Geometry & Topology 11 (2007) 473–515
 arXiv: math.GR/0511083
##### 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 $Out\left(G\right)$ either contains non-abelian free groups or is virtually nilpotent of class $\le$2. It has torsion only at finitely many primes.

One may decide algorithmically whether $Out\left(G\right)$ 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$\left(G\right)$ virtually nilpotent.

If $G$ is unimodular (virtually ${F}_{n}×ℤ$), then $Out\left(G\right)$ is commensurable with a semi-direct product ${ℤ}^{k}⋊Out\left(H\right)$ with $H$ virtually free.

##### Keywords
Baumslag–Solitar, automorphisms, graphs of groups
##### Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20E08, 20F28
##### Publication
Published: 16 March 2007
Proposed: Martin Bridson
Seconded: Walter Neumann and Wolfgang Lueck
##### Authors
 Gilbert Levitt Laboratoire de Mathématiques Nicolas Oresme UMR 6139 BP 5186 Université de Caen 14032 Caen Cedex France