The Herzog–Schönheim conjecture for finitely generated groups

Abstract

Let [Formula: see text] be a group and [Formula: see text] be subgroups of [Formula: see text] of indices [Formula: see text], respectively. In 1974, Herzog and Schönheim conjectured that if [Formula: see text], [Formula: see text], is a coset partition of [Formula: see text], then [Formula: see text] cannot be distinct. We consider the Herzog–Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define [Formula: see text] the space of coset partitions of [Formula: see text] and show [Formula: see text] is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood [Formula: see text] in [Formula: see text] such that all the partitions in [Formula: see text] satisfy also the conjecture.