## Geometry & Topology

### On the automorphism group of generalized Baumslag–Solitar groups

Gilbert Levitt

#### 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 $≤$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 $Fn×ℤ$), then $Out(G)$ is commensurable with a semi-direct product $ℤk⋊ Out(H)$ with $H$ virtually free.

#### Article information

Source
Geom. Topol., Volume 11, Number 1 (2007), 473-515.

Dates
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799838

Digital Object Identifier
doi:10.2140/gt.2007.11.473

Mathematical Reviews number (MathSciNet)
MR2302496

Zentralblatt MATH identifier
1143.20014

#### Citation

Levitt, Gilbert. On the automorphism group of generalized Baumslag–Solitar groups. Geom. Topol. 11 (2007), no. 1, 473--515. doi:10.2140/gt.2007.11.473. https://projecteuclid.org/euclid.gt/1513799838

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