1. Simplicial structures on train tracks Fedor Duzhin, Nanyang Technological University, Singapore

2. Plan of the talk • Braid groups • Crossed simplicial structure • Free groups and simplicial group structure on free groups • Combinatorial description of mapping classes • Simplicial structure on train tracks

3. Braid groups • A braid is: • n descending strands • joins {(i/(n-1),0,1)} to {(i/(n-1),0,0)} • the strands do not intersect each other • considered up to isotopy in R3 • multiplication from top to bottom • the unit braid • 1 = =

4. Artin’s presentation The braid group on n strands Bn has the following presentation: = = =

5. Braid groups on the sphere Given any space M, its n-th ordered configuration space is Obviously, the symmetric group Sn acts on F(M,n) by permuting coordinates. The braid group Bn is then The pure braid group Bn is Braid group on the sphere Pure braid group on the sphere

6. Symmetric groups Symmetric group Sn consists of bijections of an n-element set to itself Presentation with generators - transpositions Relations

7. Crossed simplicial structure The braid group is a crossed simplicial group, that is, Homomorphism tothe permutation group Face-operators Degeneracy-operators Simplicial identities Crossed simplicial relation

8. Crossed simplicial structure Face-operators are given by deleting a strand:

9. Crossed simplicial structure Degeneracy-operators are given by doubling a strand:

10. Important result Jon Berrick, Fred Cohen, Yan-Loi Wong, Jie Wu: There is a following exact sequence (actually, there are more) Here the groups of Brunnian braids are In other words, a braid is Brunnian if it becomes trivial after removing any its strand.

11. Free group Free group Fn: Generators x0,x1,…,xn-1 No relations Fn is the fundamental group of the n-punctured disk AutFn is the group of automorphisms of Fn Mapping class group consists of homotopy classes of self-homeomorphisms x0 x1 xn-1

12. Free group as a simplicial group Also, Fn admits a simplicial group structure, that is, Face-operators (group homomorphisms) Degeneracy-operators (group homomorphisms) Simplicial identities:

13. Artin’s representation Artin’s representation is obtained from considering braids as mapping classes The disk is made of rubber Punctures are holes The braid is made of wire The disk is being pushed down along the braid Theorem The braid group is isomorphic to the mapping class group of the punctured disk

14. Artin’s representation Braids and general automorphisms are applied to free words on the right • Theorem (Artin) • The Artin representation is faithful • The image of the Artin representation is the set of automorphisms given by where satisfying

15. Permutative action Theorem Generally, the braid groups act on free groups so that commute for any braid a In particular, for a pure braid a, the permutation πa is identity, so we have

16. Skeleton graphs Let S be an n-punctured disk (or, generally, a surface with n punctures and k boundary components) A skeleton graph is homotopy equivalent to the entire surface. It consists of n closed edges encircling punctures and a tree. Also, there are some natural equivalence relations. For example, one can remove a vertex of valence 1 or 2 In order to give a combinatorial description to a mapping class (that is a self-homeomorphism of the surface S considered up to homotopy fixing the boundary of the disk pointwise and the set of punctures), one first defines a skeleton graph.

17. Skeleton graphs Given a homeomorphism f:S→S, the image of a skeleton graph is some other skeleton graph. f →

18. Skeleton graphs A map of a skeleton graph G to itself occurs as follows inclusion f retraction

19. Skeleton graphs Such a map induced on skeleton graphs is not a homeomorphism For example, the following disk automorphism bar means reversed induces graph map given by Graph maps like this one are used in so called train track algorithm (M. Bestvina, M. Handel)

20. Simplicial structure on train tracks This is a current co-joint work with Jon Berrick and Jie Wu Disclaimer: it’s not train tracks we construct simplicial structure on (train tracks will not even be defined in this talk) We define a certain object called labelled skeleton graph. The set of labelled skeleton graphs is related to skeleton graph maps as Skeleton graph maps Skeleton graph mapping classes Labelled skeleton graphs Free group endomorphisms

21. Labelled skeleton graph A labelled skeleton graphs looks like Each edge is labelled by a free word Closed edges are labelled by a permutation of the free generators There are some equivalence relations

22. Thanks for your attention Simplicial structure on labelled skeleton graphs Face-operator kills a closed edge (and applies the free group face operator to all labels) face degeneracy Degeneracy-operator inserts two new edges