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We will characterize topologically conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz–Krieger algebras. [ABSTRACT FROM AUTHOR]

We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups. [ABSTRACT FROM AUTHOR]

By the work of Brodzki–Niblo–Nowak–Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence. [ABSTRACT FROM AUTHOR]