The set of connective constants of Cayley graphs contains a Cantor space

Cayley graphs are homogeneous, as well as the main characters of geometric group theory. Originating from the study of polymers, the connective constant of a homogeneous graph is the exponential growth rate of the cardinality of the set of paths of length \(n\) starting at the origin and visiting no vertex twice (or more). In this paper, we establish that the set of connective constants of Cayley graphs contains a Cantor space. In particular, this set of connective constants has the cardinality of \(\mathbb{R}\).

Electronic Communications in Probability, 22:1–4, 2017