On the automorphism group of generalized Baumslag–Solitar groups

By Gilbert Levitt. 2005

103 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/be914b45255c942546e9c5e3e71424b8ba7a2cda

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Citations in the corpus

Twisted conjugacy classes in nilpotent groups (2007, unknown): “, [7], [8], [11], [18], [23], [28], and [29]).”
Deformation spaces of trees (2006, background): “t the origin (resp. endpoint) of e j. Remark. There is a more reﬁned invariant, with values in the abstract commensurator of H. For GBS groups, it is the (signed) modular homomorphism used in [18] or [25]. As in [18], we say that D has no nontrivial integer modulus if the image of ∆ contains no inte”
Quotients and subgroups of Baumslag–Solitar groups (2013, background): “This is equivalent to G having a normal infinite cyclic subgroup, and also to G being virtually Fn × Z for some n ≥ 2 (see [12]).”
Twisted Burnside-Frobenius theory for discrete groups (2006, methodology): “[5], [30], [6], [29], [43]) while in the ‘‘opposite’’ case the twisted Burnside theorem is proved we hope that various use of Theorem 3.”
Splittings and automorphisms of relatively hyperbolic groups (2012, background): “ = a4i. Any splitting of G as a graph of inﬁnite cyclic groups is a cyclic JSJ decomposition of G [For03,GL09]. Its modular group coincides with its group of twists, and is a ﬁnite abelian group (see [Lev07]). But the JSJ deformation space of BS(2,4) is quite large [Cla09], and JSJ splittings of BS(”