The Herzog-Schonheim conjecture for finitely generated groups

TL;DR

This paper investigates the Herzog-Schönheim conjecture for free groups, introducing a combinatorial approach using covering spaces to analyze coset partitions and establish conditions under which the conjecture holds.

Contribution

It develops a new combinatorial method with covering spaces to study the conjecture for free groups and identifies conditions ensuring the conjecture's validity.

Findings

Defined a metric space of coset partitions with interesting properties

Provided sufficient conditions for the conjecture to hold in certain partitions

Established that neighborhoods around certain partitions also satisfy the 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 and develop a new combinatorial approach, using covering spaces. We define $Y$ the space of coset partitions of $F_n$ 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.