Order and minimality of some topological groups

TL;DR

This paper investigates conditions under which the group of order-preserving homeomorphisms of certain compact linearly ordered spaces is minimal, showing that for specific spaces, the group topology is uniquely determined and coincides with the Zariski and Markov topologies.

Contribution

It provides a sufficient condition for minimality of $H_+(X)$ and proves that for several classical spaces, the group is even $a$-minimal, with topologies coinciding.

Findings

The group $H_+(X)$ is minimal under the given condition.

For certain spaces, $H_+(X)$ is $a$-minimal.

Zariski, Markov, and compact-open topologies coincide on $H_+(X)$.

Abstract

A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. This paper mostly deals with the topological group $H_+(X)$ of order-preserving homeomorphisms of a compact linearly ordered connected space $X$. We provide a sufficient condition on $X$ under which the topological group $H_+(X)$ is minimal. This condition is satisfied, for example, by: the unit interval, the ordered square, the extended long line and the circle (endowed with its cyclic order). In fact, these groups are even $a$-minimal, meaning, in this setting, that the compact-open topology on $G$ is the smallest Hausdorff group topology on $G$. One of the key ideas is to verify that for such $X$ the Zariski and the Markov topologies on the group $H_+(X)$ coincide with the compact-open topology. The technique in this article is mainly based on a work of Gartside and Glyn.