Symmetries of planar growth functions. II
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- by William J. Floyd PDF
- Trans. Amer. Math. Soc. 340 (1993), 447-502 Request permission
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.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 340 (1993), 447-502
- MSC: Primary 20F05; Secondary 20F32, 57M07, 57N05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1172296-3
- MathSciNet review: 1172296