On the semigroup rank of a group

TL;DR

This paper investigates the semigroup rank of groups, establishing a dichotomy related to the group rank, and computes the semigroup rank for various classes of groups, revealing connections with torsion properties and surface orientability.

Contribution

It provides a comprehensive analysis of the semigroup rank for diverse groups, including free, abelian, nilpotent, and surface groups, highlighting new relationships and exact computations.

Findings

Semigroup rank equals group rank or group rank plus one.

For finitely generated abelian groups, torsion-free groups have semigroup rank one more than group rank.

For surface groups, semigroup rank exceeds group rank by one if and only if the surface is orientable.

Abstract

For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of relatively free groups, for any variety of groups, is computed. For a finitely generated abelian group~$G$, it is proven that $G{\rm rk}S = G{\rm rk}G+1$ if and only if $G$ is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also proven that if $M$ is a connected closed surface, then $(\pi_1(M)){\rm rk}S = (\pi_1(M)){\rm rk}G+1$ if and only if $M$ is orientable.