Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetries of planar growth functions. II

Author: William J. Floyd
Journal: Trans. Amer. Math. Soc. 340 (1993), 447-502
MSC: Primary 20F05; Secondary 20F32, 57M07, 57N05
MathSciNet review: 1172296
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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.

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Article copyright: © Copyright 1993 American Mathematical Society