Simplicial structures on train tracks

Fedor Duzhin, Nanyang Technological University, Singapore

Plan of the talk

1. Braid groups

2. Crossed simplicial structure

3. Free groups and simplicial group structure on free groups

4. Combinatorial description of mapping classes

5. Simplicial structure on train tracks

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 =

Artin’s presentationThe braid group on n strands Bn has the following presentation:

= = =

Braid groups on the sphere

The braid group Bn is then

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 pure braid group Bn is

Braid group on the sphere Pure braid group on the sphere

Symmetric groups

Symmetric group Sn consists of bijections of an n-element set to itself

Presentation with generators - transpositions

Relations

Crossed simplicial structure

The braid group is a crossed simplicial group, that is,

Homomorphism to the permutation group

Face-operatorsDegeneracy-operators

Simplicial identities Crossed simplicial relation

Crossed simplicial structure

Face-operators are given by deleting a strand:

Crossed simplicial structure

Degeneracy-operators are given by doubling a strand:

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.

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

Free group

Also, Fn admits a simplicial group structure, that is,

Free group as a simplicial group

Face-operators(group homomorphisms)

Degeneracy-operators(group homomorphisms)

Simplicial identities:

Artin’s representation

Artin’s representation is obtained from considering braids as mapping classes

The disk is made of rubberPunctures are holesThe braid is made of wireThe disk is being pushed down along the braid

Theorem The braid group is isomorphic to the mapping class group of the punctured disk

Artin’s representation

Braids and general automorphisms are applied to free words on the right

Theorem (Artin)1. The Artin representation is

faithful2. The image of the Artin

representation is the set of automorphisms given by

where

satisfying

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

Skeleton graphs

Let S be an n-punctured disk (or, generally, a surface with n punctures and k boundary components)

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.

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

Skeleton graphs

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

→f

Skeleton graphs

A map of a skeleton graph G to itself occurs as follows

inclusion

f

retraction

Skeleton graphs

Such a map induced on skeleton graphs is not a homeomorphismFor example, the following disk automorphism

induces graph map given by

bar meansreversed

Graph maps like this one are used in so calledtrain track algorithm (M. Bestvina, M. Handel)

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

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

Simplicial structure on labelled skeleton graphs

Face-operator kills a closed edge (and applies the free group face operator to all labels)

face

Degeneracy-operator inserts two new edges

degeneracy

Thanks for your attention