Trivial words in groups
by Andrew Rechnitzer
Abstract: Random walks appear at the heart of many problems in
mathematics. Perhaps one of the most famous questions is "What is the
probability that a random walk returns to its starting point?"
For a random walks on the line or the square-grid, this question can
be answered quite directly by recasting the problem as one of counting
However on more complicated graphs the problem is far from trivial. In
the setting of geometric group theory, this question is intimately
tied to the problem of "amenability" and the number of trivial words.
While amenability (and so the probability that a random walker
returns) can be decided for many groups, it remains "very open" for
Thompson's group F.
In this work, we apply numerical and enumerative methods from
statistical mechanics and combinatorics to the study of random walks
on groups and so examine the amenability of Thompson's group.
This is work together with Murray Elder, Buks van Rensburg and Thomas Wong.
No prior knowledge of group theory or statistical mechanics required....