Cutting Corners

Abstract

We define and study a class of subshifts of finite type (SFTs) defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. The main results are that for such an SFT, a locally legal pattern of convex shape is globally legal, and there is a measure that samples uniformly on all convex sets. Under suitable computability assumptions, this measure can be sampled, and legal configurations counted and enumerated, effectively and efficiently. We show by example that these subshifts need not admit a group (more generally unital magma or quasigroup) structure by shift-commuting continuous operations. Our approach to convexity is axiomatic, and only requires an abstract convex geometry that is "midpointed with respect to the shape". We construct such convex geometries on several groups, in particular all strongly polycyclic groups and free groups. We also show some other methods for sampling finite patterns, one based on orderings and one based on contructing new "independent sets" from old. We also show a link to conjectures of Gottshalk and Kaplansky.