Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings

Abstract

Let G be a finite group. We classify G-equivariant flow equivalence of non-trivial irreducible shifts of finite type in terms of (i) elementary equivalence of matrices over ZG and (ii) the conjugacy class in ZG of the group of G-weights of cycles based at a fixed vertex. In the case G = Z/2, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of E(ZG) equivalence, which involves K1(ZG). 2000 Mathematics Subject Classification 37B10 (primary), 15A21, 15A23, 15A33, 15A48, 19B28, 19M05, 20C05, 37D20, 37C80 (secondary).