Symmetries of planar growth functions. II

Abstract

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