 
Summary: Random Branched Covers of the Torus: An Exercise in Statistical
Topology
Abstract
What is the expected Euler characteristic of random branched cover of the Torus?
This quantum gravity problem motivates a number of statistical problems in group
theory and topology. We survey the techniques involved.
1 SquareTiled Surfaces
1.1 Hurwitz Encoding
We can build a surface by gluing together n squares along parallel sides. The identification
is encoded by two permutations, V, H Sn. These two permutations determine the surface
uniquely and vice versa.
Let's check this again using some algebraic topology. A tiled surface is a covering space
of the oncepunctured torus. These covering spaces can be identified with homomorphisms
from 1( ) Sn. Loops on the base lift to paths in the cover1
. The oncepunctured torus
is contractible to a wedge of two circles, so 1( ) = F2 = V, H the free group on two
generators. Homomorphisms are determined by the images of V, H in Sn.
For reference, the fundamental group of the torus is x, yxyx1
y1
since the boundary
