Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shifts

Let A be an irreducible non-permutation matrix with entries in {0,1}. We study a family ΓA,f of subgroups of the continuous full group ΓA of the one-sided topological Markov shift (XA,σA). The subgroup ΓA,f is indexed by an integer valued continuous function f on XA such that ΓA,0=ΓA the non-amenable continuous full group and ΓA,1=ΓAAF the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups ΓA,f under certain conditions on the functions f. We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups ΓA,f related to their associated groupoids and C⁎-algebras.