On continuous orbit equivalence rigidity for virtually cyclic group actions

TL;DR

This paper establishes that for minimal, topologically free actions of the infinite dihedral group on compact spaces, continuous orbit equivalence implies conjugacy, but this rigidity does not hold for some other virtually cyclic groups.

Contribution

It proves a rigidity result for infinite dihedral group actions and identifies cases where this rigidity fails for other virtually cyclic groups.

Findings

Continuous orbit equivalence implies conjugacy for dihedral group actions.

Rigidity fails for certain other virtually cyclic groups.

Provides conditions distinguishing rigid from non-rigid group actions.

Abstract

We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.