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.

A new exponential separation between quantum and classical one-way communication complexity

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string $x$, Bob receives a permutation matrix $M$, and their task is to determine whether $Mx=x$ or $Mx$ is far from $x$. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.

FOLDINGS, GRAPHS OF GROUPS AND THE MEMBERSHIP PROBLEM

We introduce a combinatorial version of Stallings–Bestvina–Feighn–Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.