# CARMA Seminar

## Thursday, 21st Nov 2013

V205, Mathematics Building

# Assoc Prof Murray Elder

(CARMA, The University of Newcastle)

# A Metropolis Markov Chain algorithm to sample trivial words and compute cogrowth in finitely generated groups; or -- F is not amenable.

In this talk I will describe an algorithm to do a random walk in the space of all words equal to the identity in a finitely presented group. We prove that the algorithm samples from a well defined distribution, and using the distribution we can find the expected value for the mean length of a trivial word. We then use this information to estimate the cogrowth of the group. We ran the algorithm on several examples -- where the cogrowth series in known exactly our results are in agreement with the exact results. Running the algorithm on Thompson's group $F$, we see behaviour consistent with the hypothesis that $F$ is not amenable.