Finitely presented nil-semigroups and aperiodic tilings SubTile 2013, CIRM, France

1. Finitely presented nil-semigroups and aperiodic tilings SubTile 2013, CIRM, France Alexey Kanel-Belov, Ilya Ivanov-Pogodaev

2. Semigroups Simple definitions Semigroup: set with an associative binary operation. Examples • set of finite words in a finite alphabet, • positive integers with addition, • ideal of a ring, etc Generators or letters: set of semigroup elements such that all other elements can be presented by them. Zero: 0 is a nil element iff 0x=x0=0 for any element x from a semigroup. Example We can define the semigroup as the set of finite words in some alphabet L. This is a free semigroup. But we can assign some relations between the letters of L, such that Word1=Word2 or Word=0. These equalities are called as defining relations. semigroup of words in the alphabet {a,b} with relations {a2=0, ba=0}. Non nil elements are: a, bn, abn for any positive integer n. Finitelely presented semigroup: a semigroup with a finite set of defining relations. Example of infinite nil semigroup Nil element: an element x such that xn=0 for some non negative integer n. semigroup of words in the alphabet {a,b,c} with infinite set of relations {X2=0} (for any finite word X). There are infinite number of non zero words. Nil semigroup: a semigroup is nil iff every element is nil.

3. Burnside problem in groups Groups: element x is periodic of period n if xn=1 for some positive integer n. Group is periodic of period n iff for any x in G: xn=1. Burnside Problem Let G be a finitely generated group of period n. Is it finite? It is true for n=1,2,3,4,6; counter examples are known: for all odd numbers greater than 660 (P.Novikov S. Adyan); even numbers : greater than several thousands (S.Ivanov, A.Lyseonok). Geometrical proof of this based on Van Campen diagram mosaics (can be found in papers of M.Gromov). Connections between group relations and tilings investigated by J.Conway in the set of papers. Remark. Nothing is known for n=5! Is there exist an infinite finitely presented periodic group? Finitely presented Burnside Conjecture

4. Finitely presented semigroups Questions In order to control defining relations in object one have to consider finitely presented objects. By other words, these objects can be presented by the finite information set. Some interesting groups, rings of semigroups can be constructed using finite number of defining relations. There are some questions here. In example V.N.Latyshev set a following problem: V.N.Latyshev Is there exist an infinite finitely presented nil-ring? This talk is devoted to the question of L.N. Shevrin. L.N Shevrin Is there exist an infinite finitely presented nil-semigroup? standard question for tilings The main point of this problem to obtain an global property (Nil-semigroup) with local tools (finite number of defining relations). Construct a nonperiodic tiling with a finite set of tiles So there are some connections between these topics

5. Finitely presented semigroups Monomial case Example There are no infinite finitely presented nil-semigroup, which defining relations are monomial. Equivalent description Let {vi} be the finite set of forbidden words. Suppose that there exists an infinite word without forbidden subwords. Then, there exists a periodic infinite word without forbidden subwords. Mathematical fantasy If we can multiply not just left-right but also upside-downside then we can use aperiodic tiling constructions. Our goal is the realization of this fantasy

6. Semigroups and tilings Three languages We want to «read» tilings as the words. So the paths on tiling would be words in the semigroups. Let us set up some connected points between semigroups and tilings. Semigroup Tiling ab=0 monomial relation abcab word, element of semigroup prohibiting boundary condition Path on mosaic (tiling) In order to construct nil-semigroup, we want to construct the set of defining relations allowing us to convert any periodic word to zero. In parralel world of tilings we can use the language of boundary conditions to obtain the property such that only non periodic tiling would be possible. possibility to convert any periodic word to zero possibility to interdict periodic tilings possibility to interdict periodic tilings using using using defining relations boundary conditions Hierarchic tiles Then, we can use hierarchic methods for tile constructing to guarantee the absence of periodic tilings. This can be obtain by theorem of Goodman-Strauss type.

7. Language of paths The most useful approach is the language of paths. Semigroup Tiling a letter in the semigroup abcaba word, element of the semigroup abdc=0 monomial relation abcd=aebd equal words Type of nodal point in the mosaic Path on mosaic (tiling), the sequence of nodal points Impossible path Equivalent paths This language can be effectively used to set up the various number of tiling conditions and in the same time can be easily translated to semigroup language.

8. Language of paths Additional ideas Every «ring» path on the tiling should be banned Only aperiodical tilings are possible So, every periodic path should be banned Good way to fix this: ban all non-geodesic paths Thus, we construct a hierarchical tiling such that all periodic paths are forbidden or can be converted to forbidden paths. So should provide a tool for considering which path are periodic. This tool is the wide possibilities of geodesic path to local changing: we can change some subpaths several times in every regular path and locally convert it into another geodesic paths with the same edge nodes. The path can «scan» the near part of tiling by local equivalents (rewriting rules) So, for any two points A and B on the tiling there exist a wide number of geodesic paths with no «choke points». The number of such geodesic paths is grow with the distance between A and B B A

9. Base graph construction Tile of level 1 Tile of level 2 This is second level of hierarhy. It is easy to provide the following levels. This is the main hieratical cutting. One tile can be cut to 6 smaller tiles. We are interesting only in topology of this tiling. Properties • every vertice has finite order; • Every path crossing some tile of level n can be locally converted such that it cross that tile by its borders; • If the path walk around 2 sides of some tile it can be locally converted to the path that walk through another two sides of that tile

10. Some additional technicalities Restitution of the mosaic’s part We can mark nodes and edges of the tiling to provide the tool of restitution.

11. Some additional technicalities Additional gluing In order to obtain the absence of choke points we use a special «additional gluing» attached to the base planar tiling Base planar tiling

12. Thank you for you attention