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
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
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