COSET ENUMERATION FOR CERTAIN INFINITELY PRESENTED GROUPS

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.

THE MANIFESTATION OF GROUP ENDS IN THE TODD–COXETER COSET ENUMERATION PROCEDURE

The issue of recognizing group properties, such as the cardinality of the group, directly from the dynamics of an incomplete coset enumeration is discussed. In particular, it is shown that the property of having two ends is recognizable in such a way. Further, sufficient conditions are given for termination of a coset enumeration with the declaration that the group under consideration has infinitely-many ends.

Double-coset enumeration algorithm for symmetrically generated groups

A double-coset enumeration algorithm for groups generated by symmetric sets of involutions together with its computer implementation is described.