Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] M. Benson, Growth series of finite extensions of $ {\mathbb{Z}^n}$ are rational, Invent. Math. 73 (1983), 251-269. MR 714092 (85e:20026)
  • [2] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), 123-148. MR 758901 (86j:20032)
  • [3] -, The growth of the closed surface groups and the compact hyperbolic Coxeter groups, unpublished manuscript.
  • [4] W. J. Floyd and S. P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), 1-29. MR 877003 (88m:22023)
  • [5] -, Symmetries of planar growth functions, Invent. Math. 93 (1988), 501-543. MR 952281 (89f:22016)
  • [6] F. R. Gantmacher, The theory of matrices, vol. 1, Chelsea, New York, 1977. MR 1657129 (99f:15001)
  • [7] P. Wagreich, The growth function of a discrete group, Proc. Conf. Algebraic Varieties with Group Actions, Lecture Notes in Math., Vol. 956, Springer, Berlin, Heidelberg, New York, 1982, pp. 125-144. MR 704992 (85k:20146)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20F05, 20F32, 57M07, 57N05

Retrieve articles in all journals with MSC: 20F05, 20F32, 57M07, 57N05

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society