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

Abstract

For a group G and a finite set A, denote by $\mathrm{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\mathrm{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\mathrm{End}(A^G)$ and $\mathrm{Aut}(A^G)$. In the first part, when G is a finite group, we give upper and lower bounds for the rank of $\mathrm{Aut}(A^G)$ 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 $\mathrm{End}(A^G)$ is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups.