A dynamical approach to Schur's Theorem

Abstract

A classical result of Schur of 1904 shows that an abstract group with finite central quotient has finite derived subgroup. Schur's Theorem has many important consequences and generalizations, which have been extensively investigated in the literature. We develop a new dynamical interpretation of Schur's Theorem for locally compact groups, using the notion of topological entropy of Adler, Konheim and McAndrew. We first consider groups with compact central quotient, introduced and called $\mathsf{Z}$-groups by Grosser and Moskowitz in the 1960s, proving that if $G$ is a connected group such that $G/Z(G)$ is compact and with continuous endomorphisms of finite topological entropy, then also $\overline{[G,G]}$ is compact and with continuous endomorphisms of finite topological entropy. The connectedness assumption is essential, since its absence allows us to construct a profinite group as counterexample. Furthermore, we study the Heisenberg groups $\mathbb{H}_n(R)$ on certain locally compact rings $R$ as a framework in which a dynamical Schur-Type Theorem persists, even though the central quotient need not be compact. In particular, we find new formulas for the $p$-rank of these Heisenberg groups.