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Coset enumeration for certain infinitely presented groups. (English) Zbl 1235.20031

Summary: We describe an algorithm that computes the index of a finitely generated subgroup in a finitely \(L\)-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely \(L\)-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group, and the Hanoi 3-group.

MSC:

20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
68W30 Symbolic computation and algebraic computation
20E07 Subgroup theorems; subgroup growth
20E28 Maximal subgroups
20-04 Software, source code, etc. for problems pertaining to group theory

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