We define a new invariant for symbolic Z2-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 Z2-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 Z2-action that it of a topological space bears to ri0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of Z2-actions, and use it to prove non-existence of certain constant-to-one factor maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have different projective fundamental groups.
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