Construction of finitely presented infinite nil-semigroup

1. Construction of finitely presented infinite nil-semigroup Ilya Ivanov-Pogodaev, Alexey Kanel-Belov

2. Semigroups Definitionsfor convenience Semigroup: a set with an associative binary operation. Example of infinite nil semigroup semigroup of words in the alphabet {a,b,c} with infinite set of relations {X2=0} (for any finite word X). There is an infinite number of non zero words. Generators or letters: a set of semigroup elements such that all other elements can be presented by products of them. Zero: 0 is a nil element in a semigroup iff 0x=x0=0 for every element x from this semigroup. We can define a semigroup by 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. Finitely presented semigroup: a semigroup with a finite set of defining relations. Nil element: an element x such that xn=0 for some positive integer n. Nil semigroup: a semigroup is nil iff every element is nil.

3. Burnside problem in groups Burnside problem in groups Let G be a finitely generated group of period n. Is it finite? It is true for n=1,2,3,4,6; counterexamples are known: for all odd numbers greater than 665 (Novikov and Adyan, recently Adyan claimed 101); even numbers: greater than several thousands (S. Ivanov, Lyseonok). Olshansky and Rips and Gromov provided geometrical methods, based on van Kampen diagram mosaics. Connections between group relations and tilings are investigated by Conway in a set of papers. Remark. Nothing is known for n=5. Finitely presented Burnside Conjecture Does there exist an infinite finitely presented periodic group? Rips proposed a conjectural example. Olshanskyand Sapir constructed non-amenable finitely presented torsion-by-cyclic groups, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, May 2003, Volume 96, Issue 1, pp 43-169 Related questions infinitely presented case: Finitely presented nil-ring Conjecture (V.Latyshev) Does there exist an infinite finitely presented nil-ring? Finitely presented nil-semigroup Conjecture (L.Shevrin and M. Sapir) Does there exist an infinite finitely presented nil-semigroup?

4. Finitely presented nil semigroup Conjecture The main question: How can one control consequences that follow from defining relations? New idea:to use the connection with tilings standard question for tilings The main problem How can one construct an aperiodic tiling with a finite set of tiles? How canone obtain a global property(nil-semigroup) using local tools (finite number of defining relations)? Three possible ways Groebner bases and Diamond lemma Penrose tiling Robinson tiling Small cancellations theory Realization of Turing Machine or Minsky Machine It is hard to construct objects in finitely presented case. The main tool is automata approach These are two similar cases, so we can use the language of tilings to write down the properties of a semigroup

5. Semigroups and tilings Here we would like to show connection between the languageof semigroups and the language of tilings: Semigroup Tiling a letter in the semigroup a word, i.e. element of the semigroup a monomial relation equal words Type of nodal point in the mosaic (tiling) a abcaba A path on mosaic (tiling), i.e. the sequence of nodal points An impossible path abdc=0 Equivalent paths abcd=aebd Using an appropriate coding of nodes and edges of a tiling we can define the corresponding semigroup. Thus, properties of the semigroup depend on the properties of the tiling. The nil-property of the semigroupcorresponds to aperiodicity of the tiling. If the tiling is ‘locally finite’ then the semigroup is finitely presented.

6. A B Semigroups and tilings: Three languages Substitution tiling It is also known that asubstitution tilingis aperiodic in general. A Consider a finite set of tiles A, B, C … Suppose that every tile of this set is subdivided into finite number of smaller tiles of same types A, B, C,… Using this substitution we can construct some tiling on the plane. A B B A In 1998 Chaim Goodman-Strauss proved that every substitution tiling can be enforced with finite matching rules (boundary conditions). So, there is a connection between languages: the language of defining relations can be translated into the language of boundary conditions, which in turn can be translated into the language of substitution tilings. Using the language of tilingsone can construct the required mosaic. possibility to forbid periodic tilings possibility to forbid periodic tilings possibility to convert any periodic word to zero using using using boundary conditions defining relations substitution tiles

7. Language of paths: Additional ideas Every path on a tiling is a word in the corresponding semigroup. Thus every periodic path should be forbidden. Every closed path on the tiling should be forbidden Only aperiodic tilings are possible So, every periodic path should be banned Good way to fix this: ban all non-geodesic paths Thus, we construct a substitution tiling so that all periodic paths can be converted to forbidden paths. So we should provide a tool to decide which paths are periodic. Thus we would like to learn how to transform paths. Local transformations c we can change some subpaths several times in every regular path and locally convert it into another geodesic path with the same endpoints. Ellipticity The path can «scan»its neighbourhood by local transformations (rewriting rules) B For every pair of points A and B on the tiling there exists a lot of geodesic paths with no «narrow places».The number of such geodesic paths grows together with the distance between A and B. A

8. Base graph construction Tile of level 1 Tile of level 2 Tile of level 3 This is the main substitution. One tile can be cut into 6 smaller tiles. We are interested only in topology of this tiling. Properties • every vertex has finite degree • Every path crossing some tile of level n can be locally transformed so that the new pathis on the boundary of the tile • If a path contains 2 sides of some tile then this path can be locally transformed to the path which contains the other two sides of that tile

9. Coding of vertices and edges CUR RU CUL U Codings of nodes L Transformation of paths on the mosaic corresponds to to transformation of codes of these paths. The codes of paths are words in the semigroup. U A B U A RU C DR L A B B B C R L A C B A C DR RU L R RD DL D C B C B Codings of incoming and outcoming edges C U A A L UL LD A C 1 B lu CDR CDL U D R We use symbols A, B, C, L, U, R, D, UL, LU, UR, RU, LD, DL, RD, DR, CDL, CDR, CUR, CUL to code all possible node types 3 Vertex of type A 2 For every node type we can name all incoming and outcoming edges ld The code of a path is a sequence of node types and edge types (with some additional technicalities)

10. lu 1 3 type A 2 ld Defining relations in the semigroup There are several types of defining relations in the semigroup: 1. Codesof sufficiently short impossible paths are equal to the nil word 1-A-4 =0 2. Codesof sufficiently short nongeodesic paths are equal to the nil word U-1-1-A-2-1-C-2-3-B =0 3. Flip relations U-1-1-A-2-1-C =U-2-1-B-3-2-C

11. Flips relations All possible transformations of paths can be obtained by repetitionof ten basic transformations.We perform these transformations on different hierarchical levels. The code of a path in some pair determines the code of the other path. We check this property for all possible cases.

12. Some additional technicalities In order to avoid narrow places we use a special «additional gluings» attached to the base planar tiling

13. Word checkby local transformations Here we show how a word of the semigroup can ‘check itself’ by some sequence of local transformationsof the path (and the corresponding code)

14. Flip defining relations (one case) There are many possibilities of how a path consisting of two edges can be situated on thecomplex. Here is one of them. For eachcase we can check that the code of one path from a pair determines the code of the other one. After this check we can show that every word either vanishes or this word is a code of some path situated on the complex. We notice that there are no periodic paths on the complex. Every code which isthe 9th degree of a word can be transformed to the nil word.

15. Basic ideas for nil-ring construction In the case of rings we have to work notwith a unique path but with a bunch of several paths (monomials). We use not only geometric coding described above but introduce additional coloringthat helps us to provide required equivalences of monomials that have the same geometric structure. Thus,forfixed geometry, there is an equivalence relation on the set of sufficiently long monomials. Our purpose is to find a coloring so that thereis a prime p such that thereare pk elements in each equivalence class.

16. Thank you for your attention.

17. Hierarchical graph Suppose that we have a finite set of graph-tiles: every one of them is a square divided into simple 4-cycles. Every side of any graph-tile contains n vertices. Finite set of graph-tiles 1U1 1U2 kU1 kU2 1Un kUn 1Ln kLn 1R1 kR1 kLn-1 1Ln-1 1R2 kR2 Graph-Tile 1 Graph-Tile k Let us call these vertices as 1Lj or (D, R, U depends on the side of the tile). Also, inner vertices of tile k are kA1.. kAm(k) 1L1 1Rn kL1 kRn 1Dn-1 1D1 kDn-1 kD1 1Dn kDn Also there are some rules describing which type of tile is used to cut every inner 4-cycle of each tile. So, we have an hierarchical rules for these tiling. Dividing structure of tiles Graph-Tile 1 1U2 1Un 1U1 every simple cycle is 4-cycle 1A2 T1 1Ln T5 1R1 1A1 Example 4-cycle is divided according the corresponding tile structure 1Ln-1 T5 1R2 T4 1A4 T T T T T 1A3 T 1Am(k) dividing tile can be turned to any side 1L1 1Rn T6 1Dn-1 1Dn 1D1

18. A1 A3 A2 A4 Ai Tile V Tile U Ai URiVLn+1-i Types of vertices Let us consider which types of vertices we can see in the graph. Black White The vertices on the border of two tiles are white. These vertices can be described as combinations of side vertices Li Di Ri Ui for all types of tiles. If the right side of U tile meet the left side of V tile, then there are RiLn+1-i white vertices on the border. UR1VLn We call inner Ai vertices as black. Tile V Tile U UR2VLn-1 URnVL1 Local structure of a black vertex Consider some Ai vertex in the tile T. Let us fix the border edges of the tiles from which our tile consists of (the next level of hierarchy). Some of them are Ai out-edges. Let us call them Main edges for Ai. Maybe Ai have some more edges (which can appear if we construct the next level of hierarchy). Let us call them Secondary edges for Ai. Local structure of a white vertex Consider some vertex URiVLn+1-i on the border of U and V tiles. There are two out-edges situated also on the border. Let us call them Main edges for URiVLn+1-I Also, the vertex can have other out-edges (into U or V). Let us call them Secondary edges. Finite degree condition for the graph There exists a global constant D, such that the degrees of all vertices is lower than D. This is the special condition for the tile composition rules.

19. Tile V Tile U URiVLn+1-i Y Z W Additional parameters (shades) If we know the vertices on the path, we know some local structure of a graph. But we need additional information about this structure. Thus, we consider some additional parameters for white vertices of our graph. Second parameter (pair of «bosses») First parameter (edge code) For any tile we can code all decomposition edges of it. Let the code of the edge be the first parameter of all white vertices on the edge except the ends (X,Y). X There are two vertices X,Y in the opposite ends of the border edge. Let the combination of (X,Y) types be the second parameter for all white vertices on the border edge. X V V U Y Y Finally, the «vertex» isa combination of vertex type (Ai, or LiRj, or UiLi etc), first and second parameters. So, if we know some white vertex X, we know that biggest tiles meets here, where in bigger tile these two tiles are situated, and which vertices are the «bosses» of this edge. Path reading Consider the path on graph. Then the path walks through some vertex, we can note the edges wherefrom this path comes and where it goes. So we can code all incoming and outcoming edges, and write a finite set of all paths on our graph with length lower than some global constant N. If we know nothing about the graph, but have the full code of the path, we can examine it and find some information about local structure. For example, if we walk through some main edges and then turn to secondary edge, this means that the path went into the tile (and we know which tile is it). X

20. B2 Z Y Tile V Tile U W X B1 If there are no forbidden paths Now we can consider the class of graphs with declared set of vertices with the condition that there are no forbidden paths. Consider some two bosses B1 and B2 and long enough path between them consists of white only vertices and walks on main edges only. Next to B1 vertex should have a type G1Hn, (where G,H can be L,R,U,D types). Note that indexes are (1,n). Next vertex has indexes (2,n-1), and so on. So we can see that on any n+1 sector there are n vertices which indexes are increasing by 1, and one vertex have some other pair of indexes. Let us call such vertices high-leveled, and the other 1-leveled. Ring path lemma If high leveled X vertex has the indexes (i,n+1-i), for I<n, then the next high-leveled vertex Y has the indexes (i+2,n-i). In hierarchical graph there is a path to some black vertex W. So in our situation we can walk from X to the edge on the way to W. It is easy to see that first black vertex we will meet should have the same type as W. Otherwise we can choose a forbidden path. From W, we can turn to the edge on the way to Z (as in hierarchical graph) walk to first black vertex. It can be only Z. From Z we can walk to Y (as in hierarchical graph). So, Y in our graph should have the same type as in hierarchical graph, so Y has the indexes (i+2,n-i).

21. Introduction This talk is devoted to the method for control an infinite structures by some local rules. We consider a hierarchical graph S, then fix the finite set of vertices types. Let us call a finite sequence of graph vertices as a path. Question Is it possible to write a finite set of forbidden paths such that the following condition holds: If an infinite graph with declared set of possible vertices contains no forbidden paths then any local part of it is a subgraph of our hierarchical graph S. The answer for this question is yes. We can fix some large enough constant N and consider all paths with length lower than N, which are not exist in the graph S. The structure of such hierarchical graph can be used for construction of finitely presented semigroups. For example, this method used for the construction of finitely presented nil-semigroup.