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


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.

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