Geometry & Topology

On the automorphism group of generalized Baumslag–Solitar groups

Gilbert Levitt

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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×), then Out(G) is commensurable with a semi-direct product k Out(H) with H virtually free.

Article information

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

First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20E08: Groups acting on trees [See also 20F65] 20F28: Automorphism groups of groups [See also 20E36]

Baumslag–Solitar automorphisms graphs of groups


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.

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