Connective Constants on Cayley Graphs

Abstract

For a transitive infinite connected graph $G$, let $\mu (G)$ be its connective constant. Denote by $\mathcal{G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of other generators. Assume $G\in \mathcal{G}$ is a Cayley graph of a finitely presented group, and Cayley graph sequence $\{G_n{\}}_{n=1}^{\infty }\subset \mathcal{G}$ converges locally to $G.$ Then $\mu (G_n)$ converges to $\mu (G)$ as $n\to \infty .$ This confirms partially a conjecture raised by Benjamini [2013. {\it Coarse geometry and randomness.} Lect. Notes Math. {\bf 2100}. Springer.] that connective constant is continuous with respect to local convergence of infinite transitive connected graphs.