Tiling Problems on Baumslag-Solitar groups.

Abstract

We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups

The large scale geometry of strongly aperiodic subshifts of finite type

“…Berger's result shows that Z 2 has undecidable domino problem, whereas we have remarked that Z is known to have decidable domino problem. Aubrun and Kari have shown that the Baumslag Solitar groups have undecidable domino problem [1]. Ballier and Stein [2], building on results from several authors [19][20] [16] [15], observe that every virtually free group has decidable domino problem, and conjecture that these are the only such groups.…”

“…Berger showed that Z 2 has a strongly aperiodic subshift of finite type [3]. Many other groups are known to admit such subshifts, including higher rank free abelian groups [11], solvable Baumslag Solitar groups [1], the integral Heisenberg group [10], cocompact lattices in higher rank simple Lie groups [18], and the direct product of Thompson's group T with Z [14]. Forthcoming work of the author and Goodman-Strauss will show that surface groups also have such subshifts [7].…”

“…Proof. If G has undecidable word problem, then H also has undecidable word problem [12, Theorem 2.2.5], and therefore G and H both have undecidable domino problem as it is well known (see for instance the Introduction of [1]) that a group with undecidable word problem also has undecidable domino problem. Consequently, we may assume without loss of generality that G and H have decidable word problem.…”

Non-cooperative Algorithms in Self-assembly

“…the sequence (y P i ) i of its y-coordinates, or the sequence (x P i ) i of its x-coordinates is monotonic), and repeats a tile type, is pumpable. Therefore, the main ingredient of efficient paths is non-monotonicity: we call a vertical cave (respectively horizontal cave) a part of a path P between two indices i and j, such that (1)…”

“…In this section, we show that we are far from understanding these relations, and begin a broader exploration of the influence of geometry. In Wang tilings, geometries that have been considered previously include the hyperbolic plane [12,16] and Cayley graphs of Baumslag-Solitar groups [1,2].…”