Some examples of monothetic groups

TLDR

Section 0 surveys related results already existing in the literature. The paper discusses the finest group topology on ℤ making the even integers converge to zero and presents a complete metrizable, totally disconnected monothetic group with no nontrivial continuous characters. The authors construct the example by starting with a complete metrizable totally disconnected monothetic group, factoring out a dense discrete subgroup of its Bohr compactification, and use a 2‑adic coordinate trick from Section 1 to establish discreteness. They prove that ℤ is a complete topological group under this topology, with a self‑contained proof.

Abstract

In this paper we discuss in Section 1 the finest group topology on Z zuch that {2n}nN is a sequence converging to zero. We prove that Z is a complete topological group with respect to that topology. The proof is almost entirely self-contained. In section 2 we give an example of a monothetic group (that is a topological group which contains a dense copy of Z) which is complete metrizable, totally disconnected and which has no continuous characters except 0. The example is constructed by starting with a complete meterizable totally disconnected monothetic group and factoring out a discrete subgroup which is dense in the Bohr compactification. The main difficulty here is in proving discreteness. Section 2 relies heaviliy on a trick developed in Section 1, namely the use of a kind of coordinates for 2-adic numbers. Section 0 contains a survey of related results already existing in the literature.