Full groups of one-sided topological Markov shifts

Let (X (A) , sigma (A) ) be the right one-sided topological Markov shift for an irreducible matrix with entries in {0, 1}, and I" (A) the continuous full group of (X (A) , sigma (A) ). For two irreducible matrices A and B with entries in {0, 1} satisfying condition (I), it is proved that for a group isomorphism alpha from I" (A) to I" (B) , there exists a homeomorphism h from X (A) to X (B) such that alpha(gamma) = h au < gamma au < h (-1) for gamma a I" (A) and hence h au < I" (A) au < h (-1) = I" (B) . As a result, we know that the continuous full groups I" (A) and I" (B) are isomorphic as abstract groups if and only if their one-sided topological Markov shifts (X (A) , sigma (A) ) and (X (B) , sigma (B) ) are continuously orbit equivalent.