On the automorphism group of generalized Baumslag–Solitar groups
On the automorphism group of generalized Baumslag–Solitar groups
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广义鲍姆斯拉格-索利塔群的自同构群
DOI：
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发表时间：
2005
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通讯作者：
G. Levitt
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文献类型：
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作者：
G. Levitt
广义Baumslag-Solitar群（GBS群）是一个作用在树上的群G$，它的所有边和顶点稳定子都是无限循环的.我们证明了Out（G）要么包含非交换自由群，要么是至多2类的虚幂零群。它只有在1000个素数时才有挠。人们可以通过算法来决定Out（G）是否是虚幂零的。如果是的话，我们就可以决定它实际上是阿贝尔的，还是可交换生成的。同构问题是可解的GBS群与出（G）几乎幂零。如果$G$是幺模的（实际上是$F_n \times Z$），则Out（G）可以与半直积$Z^k \rtimes Out（H）$交换，其中$H$实际上是自由的。
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.