On the minimal number of generators of endomorphism monoids of full shifts

TL;DR

This paper investigates the minimal number of generators needed for the monoid of shift-commuting maps on full shifts over groups, providing bounds for finite groups and showing non-finite generation for certain infinite groups.

Contribution

It establishes bounds on the rank of automorphism groups for finite groups and proves that endomorphism monoids are not finitely generated for many infinite groups.

Findings

Bounds for automorphism group rank based on subgroup conjugacy classes

Endomorphism monoids are not finitely generated for groups with infinite descending chains of finite index normal subgroups

Applicable to classes like residually finite and locally graded groups

Abstract

For a group $G$ and a finite set $A$, denote by $\text{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\text{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{End}(A^G)$ and $\text{Aut}(A^G)$. In the first part, when $G$ is a finite group, we give upper and lower bounds for the rank of $\text{Aut}(A^G)$ in terms of the number of conjugacy classes of subgroups of $G$. In the second part, we apply our bounds to show that if $G$ has an infinite descending chain of normal subgroups of finite index, then $\text{End}(A^G)$ is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.