The finite tiling problem is undecidable in the hyperbolic plane

In this paper, we consider the finite tiling problem which was proved undecidable in the Euclidean plane by Jarkko Kari in 1994. Here, we prove that the same problem for the hyperbolic plane is also undecidable

About the domino problem in the hyperbolic plane, a new solution: complement

In this paper, we complete the construction of paper arXiv:cs.CG/0701096v2. Together with the proof contained in arXiv:cs.CG/0701096v2, this paper definitely proves that the general problem of tiling the hyperbolic plane with {\it \`a la} Wang tiles is undecidable.Comment: 20 page

Around the Domino Problem – Combinatorial Structures and Algebraic Tools

Given a finite set of square tiles, the domino problem is the question of whether is it possible to tile the plane using these tiles. This problem is known to be undecidable in the planar case, and is strongly linked to the question of the periodicity of the tiling. In this thesis we look at this problem in two different ways: first, we look at the particular case of low complexity tilings and second we generalize it to more general structures than the plane, groups. A tiling of the plane is said of low complexity if there are at most mn rectangles of size m × n appearing in it. Nivat conjectured in 1997 that any such tiling must be periodic, with the consequence that the domino problem would be decidable for low complexity tilings. Using algebraic tools introduced by Kari and Szabados, we prove a generalized version of Nivat’s conjecture for a particular class of tilings (a subclass of what is called of algebraic subshifts). We also manage to prove that Nivat’s conjecture holds for uniformly recurrent tilings, with the consequence that the domino problem is indeed decidable for low-complexity tilings. The domino problem can be formulated in the more general context of Cayley graphs of groups. In this thesis, we develop new techniques allowing to relate the Cayley graph of some groups with graphs of substitutions on words. A first technique allows us to show that there exists both strongly periodic and weakly-but-not-strongly aperiodic tilings of the Baumslag-Solitar groups BS(1, n). A second technique is used to show that the domino problem is undecidable for surface groups. Which provides yet another class of groups verifying the conjecture saying that the domino problem of a group is decidable if and only if the group is virtually free

Groups, Graphs, Languages, Automata, Games and Second-order Monadic Logic

In this paper we survey some surprising connections between group theory, the theory of automata and formal languages, the theory of ends, infinite games of perfect information, and monadic second-order logic

Combinatorics of Pisot Substitutions

Siirretty Doriast