The space of coset partitions of $F_n$ and Herzog-Sch\"onheim conjecture

Abstract

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim conjectured that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Sch\"onheim conjecture for free groups of finite rank. We define $Y$ the space of coset partitions of $F_n$ and show $Y$ is a metric space with interesting properties. In a previous paper, we gave some sufficient conditions on the coset partition of $F_n$ that ensure the conjecture is satisfied. Here, we show that each coset partition of $F_n$, which satisfies one of these conditions, has a neighborhood $U$ in $Y$ such that all the partitions in $U$ satisfy also the conjecture.