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

TL;DR

This paper explores the structure of coset partitions in free groups and demonstrates that certain partitions satisfying Herzog-Schönheim conjecture conditions have neighborhoods where all partitions also satisfy the conjecture.

Contribution

It introduces a metric space framework for coset partitions of free groups and proves stability of Herzog-Schönheim conjecture conditions within this space.

Findings

The space of coset partitions of $F_n$ is a metric space with notable properties.

Partitions satisfying certain conditions have neighborhoods where all partitions satisfy the conjecture.

The results support the conjecture's validity in a neighborhood-based setting.

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.