Height functions for amenable groups

Geoffrey R. Grimmett, Zhongyang Li

Published 2015-10-29Version 1

The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. Various properties of connective constants depend on the existence of so-called graph height functions, namely: (i) whether $\mu(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $\mu(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $\mu(G)$ to a given degree of accuracy. Graph height functions are explored here on Cayley graphs of infinite, finitely presented groups $\Gamma$, in which context they are related to integer-valued surjective homomorphisms on the finite-index subgroups of $\Gamma$. We prove that the Cayley graphs of infinite, finitely generated, elementary amenable groups support graph height functions, which are in addition harmonic. In contrast, we show that the Cayley graph of the first Grichorchuk group, which is amenable but not elementary amenable, does not have a graph height function. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group. This work extends the set of groups for which graph height functions are known to exist, and resolves in the negative an open question concerning the existence of height functions on general transitive graphs.