Representation of the mapping class group

Description

This article is mostly due to lit_koberdaAsymptoticLinearityMapping2010. We give a few known facts on the representation of mapping class groups.

Properties

Problem: Linearity

It is unknown whether the Mapping class group is linear or not (meaning if it permits a faithful representation in a finite-dimensional space). However, Braid groups and mapping class groups of surfaces of genus $2$ are linear.

Non-triviality action of characteristic fundamental subgroups on the homology of the characteristic cover

Let $\varphi \in MC{G}^1(S)$ and $S_{\Gamma }$ be a Characteristic cover with a finite Characteristic quotient $\Gamma ={\pi }_1(S)/{\pi }_1(S_{\Gamma })$. Think of the subgroup ${\pi }_1(S)\subseteq {\pi }_1(S_{\Gamma })$ as the curve classes which can be lifted to the cover and the quotient $\Gamma ={\pi }_1(S)/{\pi }_1(S_{\Gamma })$ as the curves classes which cant be lifted. If $\varphi \in Aut(\Gamma )$ (we use the identification $MC{G}^1(S)=Aut({\pi }_1(S,\ast ))$ for a base point $\ast$) is non-trivial, then $\varphi$ induces a non-trivial action on $H_1({\Sigma }_{\Gamma ,}\mathbb{Z})$. *Elements in $Aut(\Gamma )$ move non-lifting curves to non-lifting curves. The vectors in $H_1({\Sigma }_{\Gamma ,}\mathbb{Z})$ are class of curves. *

We can apply the above theorem repeatedly. Take a cover $S^{\prime }$ of $S$. Now take a characteristic cover $S_{\Gamma }^{\prime }$ of the cover. The nontrivial elements of the quotient $\Gamma$ are mapping classes and act non-trivially on the homology of $S_{\Gamma }^{\prime }$. We will get a sequence of quotients ${\Gamma }_i$ where $Aut({\Gamma }_i)$ exhaust the elements in $MCG(S)$.

Let an exhausting sequence of covers be a sequence of covers whose fundamental groups converge to $\{id\}$. Call a cover solvable, nilpotent or $p$-cover if its fundamental group is a solvable, nilpotent or $p$-subgroup. We have the following theorem:

Every mapping class acts non-trivially on some cover homology.

Let $\{{\Sigma }_i\}$ be a sequence of exhausting, finite characteristic covers of $\Sigma$. Then for a $\varphi \in MCG(S)$ there is a $i$ such that $\varphi$ acts non-trivially on $H({\Sigma }_i,\mathbb{Z})$. Furthermore we can assume that $\varphi$ acts nontrivially on the homology of a solvable, nilpotent or even $p$-cover for any prime $p$.

• What is the relationship to solvable, nilpotent or $p$-covers? Does the theorem say that we can always find a cover which is big enough for $\varphi$ to act non-trivially and then we find a solvable, nilpotent or a $p$-subgroup inside the fumdamental group?

Next we want to show that the action of the mapping class group on the homology of the cover is “outer” where as the action of a deck transformation is “inner”. Meaning they do not coinside.

The action of $MC{G}^1$ and $\Gamma$ on the cover do not coincide

Let $S$ be a surface with finite characteristic cover ${\Sigma }^{\prime }\to \Sigma$. Suppose the action of $\varphi \in MC{G}^1(S)$ on $H_1({\Sigma }^{\prime },\mathbb{Z})$ coincides with a Decktransformation of the cover, with base point $\ast$. (Deck transformations exchange curves on the cover, inducing an action on homology). Then $\varphi$ induces an inner automorphism of ${\pi }_1(\Sigma )$. Meaning, if we forget the basepoint $\ast$ then the element becomes the trivial element in $MCG(S)$

• What is this used for?

Examples

Representation by Homology

A simple representation is obtained by studying the action of $MCG(\Gamma )$ on the homology. This gives us a representation $MCG(\Gamma )\to GL(n,\mathbb{Z})\cong Aut(H_1(\Sigma ,\mathbb{Z}))$ The kernel of this homomorphism is the complicated Torelli Group.

Representation from Homology of Covers

We can add one more step in the above construction. If we lift $\Gamma$ to a cover $\tilde{\Gamma }$ and look at a mapping class that lifts to $\tilde{\Gamma }$ as well. This mapping class then acts on the slightly bigger homology of $\tilde{\Gamma }$, decreasing the size of the torelli group. By doing so on higher and higher covers, we can find a faithful action on an infinite-dimensional vector space.