The Projective Fundamental Group of a Z 2 -shift Projective Fundamental Group of a Z 2 -shift)

We deene a new invariant for symbolic Z 2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the Z 2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties of the Z 2-action that 1 of a topological space bears to 0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of Z 2-actions, and use it to prove nonexistence of certain kinds of constant-to-one maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have diierent projective fundamental groups.