On the topological entropy of nilpotent groups of finite rank - Waka, Olwethu (Department of Mathematics and Applied Mathematics, 2025)
The present PhD thesis deals with some finiteness conditions on the topological entropy of continuous endomorphims of periodic locally compact Heisenberg p-groups (p prime). These are relevant models of locally compact nilpotent groups and their structure may be described very well by semidirect products. Firstly, we consider the class of locally compact abelian groups that are compactly generated; we recognize a relevant subclass among these, namely the slender groups, and observe that slender groups have very small values of topological entropy for their endomorphisms. Then we investigate a more general case of nilpotent periodic locally compact p-groups, reducing the computations to maximal p-subgroups. The role of p-groups becomes somehow fundamental, so we focus on locally compact Heisenberg p-groups which are studied especially on the field Qp of p-adic rationals and on the ring Zp of p-adic integers.