# Trivial words in groups

*by Andrew Rechnitzer*

*Institution:*University of British Columbia, Canada

*Date: Tue 16th April 2013*

*Time: 12:00 PM*

*Location: Russell Love Theatre, Richard Berry Building*

*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

loops.

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....