# Covering groups of minimal exponent

## TL;DR

This paper explores how certain group presentations via free products of cyclic groups relate to covering groups with minimal exponent, linking algebraic and topological structures through simplicial complexes and surface coverings.

## Contribution

It establishes conditions under which covering groups have minimal exponent, connecting algebraic group presentations with topological surface coverings.

## Key findings

- Covering groups with minimal exponent are characterized by preserving generator orders.
- The correspondence between group presentations and surface coverings is formalized.
- A link between algebraic and topological structures via simplicial complexes is demonstrated.

## Abstract

Presenting a finite group by a free product of finite cyclic groups the Hopf formula for the Schur multiplier affords also a covering group, and this has minimal exponent provided that the order of the generators is preserved. This condition corresponds to a covering projection between simplicial complexes, and so a presentation by a Fuchsian group corresponds to a covering projection between compact surfaces.