The Herzog-Schönheim conjecture for finitely generated groups

Let G be a group and H1,..,Hs be subgroups of G of indices d1,..,ds, respectively. In 1974, Herzog and Schönheim conjectured that if {Hiαi}i=1i=s, αi G, is a coset partition of G, then d1,..,ds 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 Y the space of coset partitions of Fn and show Y 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 U in Y such that all the partitions in U satisfy also the conjecture.