On locally compact groups with zero

TL;DR

This paper investigates algebraic properties of groups that determine whether all locally compact shift-continuous topologies with an adjoined zero are either compact or discrete, introducing new classes called electorally flexible and stable groups.

Contribution

It introduces the concepts of electorally flexible and stable groups, establishing their properties and their influence on the topology of groups with adjoined zero.

Findings

Electorally flexible groups include groups with an infinite cyclic subgroup of infinite index.

Uncountable commutative groups are electorally flexible.

Countable locally finite groups are electorally stable.

Abstract

We study algebraic properties on a group G such that if the discrete group G has these properties then every locally compact shift continuous topology on G with adjoined zero is either compact, or discrete. We introduce electorally flexible and electorally stable groups and establish their properties. In particular, we prove that every group with an infinite cyclic subgroup of an infinite index and every uncountable commutative group are electorally flexible, and show that every countable locally finite group is electorally stable. The main result of the paper is the following: if G is a discrete electorally flexible group then every Hausdorff locally compact shift-continuous topology on G with adjoined zero is either compact, or discrete. Also, we construct a non-discrete non-compact Hausdorff locally compact shift-continuous topology on any discrete virtually cyclic group (and hence on a electorally stable group) G with adjoined zero.