Generalized Baumslag-Solitar groups: rank and finite index subgroups
Generalized Baumslag-Solitar groups: rank and finite index subgroups
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广义 Baumslag-Solitar 群：秩和有限指数子群
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发表时间：
2013
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通讯作者：
G. Levitt
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作者：
G. Levitt
广义Baumslag-Solitar（GBS）群是作用在具有无限循环边和顶点稳定子的树上的一个生成群。给出了如何有效地确定GBS群的秩（生成集的最小基数），从而可以计算自由群$F_n$的有限阶外自同构的映射环面的秩。我们还证明了GBS群G的有限指数子群的秩不可能小于G的秩。我们确定哪些GBS群是大的（一些有限指数子群映射到F_2 $上），并且我们解决了在特定的GBS群族中的可同构性问题（确定两个群是否具有同构的有限指数子群）。
A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite order outer automorphism of a free group $F_n$. We also show that the rank of a finite index subgroup of a GBS group G cannot be smaller than the rank of G. We determine which GBS groups are large (some finite index subgroup maps onto $F_2$), and we solve the commensurability problem (deciding whether two groups have isomorphic finite index subgroups) in a particular family of GBS groups.