New algorithms and new methods of analysis for Eulerian orientations and the amenability of Thompson's group F
by Tony Guttmann
Abstract: This is the second of two talks concerning two enumeration problems and some associated problems. The first problem is to count the number of rooted, planar, Eulerian orientations with exactly $n$ edges. This has close ties to statistical mechanics, as it can be seen as a question about orientations on a random lattice, and the problem restricted to 4-valent maps is exactly the six vertex model on a random lattice.
The second problem comes from geometric group theory. It asks for the number of walks of length $n$ in the Cayley graph of Thompson's group $F$ return to the initial vertex. The interest in this question primarily stems from the fact that the limiting behaviour of these numbers is known to determine whether or not the group (in this case $F$) is amenable.
In lieu of an exact solution to either problem, we have developed efficient algorithms to calculate the exact values of these numbers for as large a value of $n$ as we could. We will describe the ideas behind each of these algorithms as well as their efficiency and results.
The nature of the solutions required the development of new techniques in series analysis, and we will describe these techniques and show how they gave rise to important new conjectures for both problems.