Order and minimality of some topological groups

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 [21].