Proceedings of Symposia in Pure Mathematics

Volume 74, 2006

Fou r Q u e s t i o n s a b o u t M a p p i n g Clas s G r o u p s

Mladen Bestvina

In this note I will present my four favorite questions about mapping class

groups. The first two are particularly dear to my heart and I frequently ponder

some of their aspects.

1. Systoles and rational cohomology

Let Tg denote the Teichmuller space of marked complete hyperbolic structures

on the surface Sg of genus g, and let

Mg = Tg/MCG(Sg)

denote the quotient by the mapping class group (thus Mg is the space of unmarked

hyperbolic structures). To each hyperbolic surface E G Mg we can associate the

length L(E) of a shortest nontrivial closed geodesic in E. Such a geodesic is called

a systole. This gives us a continuous function

L: Mg -» (0,oo)

EXERCISE 1. Show that

• L attains a maximum \xg.

• Every set of the form L-1([e, /j,g]) (e 0) is compact.

In other words, the function

* := -log L : Mg — [-log(ng), oo)

is a proper function. We would like to regard ^ as a kind of a Morse function

on Mg in order to study its topology. Of course, L is not even smooth, but that

shouldn't stop us. (To see that L is not smooth imagine a smooth arc in Mg along

which the length of a curve a has positive derivative and the length of a curve (3 has

negative derivative. Suppose that the lengths of a and /3 are equal at an interior

point p of the arc, and that a is a systole on one side of this point and (3 on the

other. Then L restricted to this arc fails to be smooth at p. For g = 1 one can see

that L is not smooth by an explicit calculation - see below.)

We want to examine the change in topology of the sublevel sets

tf-'a-MM*),*])

as t passes through the "critical values" of \P.

It is a theorem of Akrout [Akr03] that \£ o q is a topological Morse function

on Tg where q : Tg —• Mg is the quotient map. Recall that a point x £ X

©2006 Mladen Bestvina

3

http://dx.doi.org/10.1090/pspum/074/2264129