An Algebraic Obstruction to Isomorphism of Markov Shifts with Group Alphabets

Given a compact group G , a standard construction of a Z 2 Markov shift ∑ G 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 ∑ G . We show that if G is sufficiently non‐abelian (for instance A 5 , PSL 2 (F 7 ) or a Suzuki simple group) and H is any abelian group with ∣ H ∣ = ∣ G ∣, then ∑ G and ∑ H are not isomorphic. Thus the cardinality of G is seen to be necessary but not sufficient to determine the measurable structure of ∑ G .