# 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.

## Key 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.