Symmetries of Planar Growth Functions. II

Abstract

Let $G$ be a finitely generated group, and let $\Sigma$ be a finite generating set of $G$. The growth function of $(G,\Sigma )$ is the generating function $f(z) = \sum \nolimits _{n = 0}^\infty {{a_n}{z^n}}$, where ${a_n}$ is the number of elements of $G$ with word length $n$ in $\Sigma$. Suppose that $G$ is a cocompact group of isometries of Euclidean space ${\mathbb {E}^2}$ or hyperbolic space ${\mathbb {H}^2}$, and that $D$ is a fundamental polygon for the action of $G$. The full geometric generating set for $(G,D)$ is $\{ g \in G:g \ne 1$ and $gD \cap D \ne \emptyset \}$. In this paper the recursive structure for the growth function of $(G,\Sigma )$ is computed, and it is proved that the growth function $f$ is reciprocal $(f(z) = f(1/z))$ except for some exceptional cases when $D$ has three, four, or five sides.