An Algebraic Obstruction to Isomorphism of Markov Shifts with Group Alphabets

Given a compact group G, a standard construction of a Z2 Markov shift SG with alphabet G is described. The cardinality of G (if G is finite) or the topological dimension of G (if G is a torus) is shown to be an invariant of measurable isomorphism for SG. We show that if G is sufficiently non-abelian (for instance A5, PSL2(F7), or a Suzuki simple group) and H is any abelian group with |H| = |G|, then SG and SH are not isomorphic. Thus the cardinality of G is seen to be necessary but not sufficient to determine the measurable structure of SG.