On the interconnectedness, via random walks, of cogrowth rates and the Følner function

Amenability can be characterised in many ways, and many of these characterisations employ limits. This thesis investigates the connection between two such characterisations of amenability; the cogrowth function, and Følner sequences. The first part of the thesis concerns recent numerical results of Elder, Rechnitzer and van Rensburg. Objections to their work on Thompson’s group F have been made on the basis of Moore’s result about the Følner function for F. We introduce a function ℛ: ℕ ⇾ ℝ which quantifies the rate of convergence of the cogrowth function to its asymptotic growth rate, in analogy to the Følner function. Growth properties of this function are shown to be capable of compromising the results of Elder, Rechnitzer and van Rensburg. Furthermore, evidence is given suggesting a correlation between these growth properties and those of the Følner function. We then modify the method proposed by Elder, Rechnitzer and van Rensburg to compute surprisingly accurate estimates for initial values of the cogrowth function, which shows that the method still gives useful information even if it cannot predict asymptotic behaviour accurately. In the second part of the thesis we investigate more generally the connections between cogrowth and Følner sequences. Random walk distributions are identified as a natural intermediary, and it is proved that a specific Følner sequence can be constructed directly from a random walk distribution. Based on the properties of the corresponding sets arising from random walks on non-amenable groups we conjecture a formulation of the amenable radical of a finitely generated non-amenable group arising from the distribution of a random walk.