Sub-dominant Cogrowth Behavior and the Viability of Deciding Amenability Numerically
Murray Elder, Cameron Rogers
We critically analyse a recent numerical method due to the first author, Rechnitzer and van Rensburg, which attempts to detect amenability in a finitely generated group by numerically estimating its asymptotic cogrowth rate. We identify two potential sources of error. We then propose a modification of the method that enables it to easily compute surprisingly accurate estimates for initial terms of the cogrowth sequence. * Research supported by Australian Research Council grant FT110100178 We
... w that, in the ERR algorithm, estimates of the asymptotic cogrowth rate are compromised by sub-dominant behaviour in the reduced-cogrowth function. However, even though sub-dominant behaviour in the cogrowth function may interfere with estimates of the asymptotic growth rate, the ERR algorithm can still be used to estimate other properties of the cogrowth function to high levels of accuracy. In particular we are able re-purpose the algorithm to quickly estimate initial values of the cogrowth function even for groups for which the determination of the asymptotic cogrowth rate is not possible (for example finitely generated groups with unsolvable word problem). The present work started out as an independent verification by the second author of the experimental results in , as part of his PhD research at the University of Newcastle. More details can be found in  . The article is organised as follows. In Section 2 we give the necessary background on amenability, random walks and cogrowth. In Section 3 a function quantifying the sub-dominant properties of the reduced-cogrowth function is defined. In Section 4 the ERR algorithm is summarised, followed by an analysis of two types of pathological behaviour in Section 5. The first of these is easily handled, while the second is shown to depend on sub-dominant terms in the reduced-cogrowth function. In Section 6 the ERR method is modified to provide estimates of initial cogrowth values. Using this the first 2000 terms for the cogrowth function of Thompson's group F are estimated. Preliminaries The following characterisation of amenability is due to Grigorchuk  and Cohen . A shorter proof of the equivalence of this criteria with amenability was provided by Szwarc . Definition 2.1. Let G be a finitely generated non-free group with symmetric generating set S. Let c n denote the number of freely reduced words of length n over S which are equal to the identity in G. Then G is amenable if and only if lim sup n→∞ c 1/n n = |S| − 1. Equivalently, let d n denote the number of words (reduced and unreduced) of length n over S which are equal to the identity. Then G is amenable if and only if lim sup n→∞ d 1/n n = |S|. The function n → c n is called the reduced-cogrowth function for G with respect to S, and n → d n the cogrowth function. Kesten's criteria for amenability is given in terms of the probability of a random walk on the group returning to its starting point.