On the automorphism group of generalized Baumslag–Solitar groups

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...

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Bibliographic Details
Published in:Geometry & topology 2007-03, Vol.11 (1), p.473-515
Main Author: Levitt, Gilbert
Format: Article
Language:English
Subjects:
Geometric Topology
Group Theory
Mathematics
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Summary: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 $Z^k \rtimes Out(H)$ with $H$ virtually free.
ISSN:1465-3060
1364-0380
DOI:10.2140/gt.2007.11.473