Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On Rationality of the Cogrowth Series

Author: Dmitri Kouksov
Journal: Proc. Amer. Math. Soc. 126 (1998), 2845-2847
MSC (1991): Primary 20F05, 20P05, 05C38
MathSciNet review: 1487319
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The cogrowth series of a group $G$ depends on the presentation of the group. We show that the cogrowth series of a non-empty presentation is a rational function not equal to 1 if and only if $G$ is finite. Except for the trivial group, this property is independent of presentation.

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

  • 1. J. M. Cohen Cogrowth and Amenability of Discrete Groups, J. of Funct. Anal. 48 (1982) 301-309. MR 85e:43004
  • 2. R. I. Grigorchuk Symmetrical random walks on discrete groups. In: Multicomponent Random Systems, ed. by R. L. Dobrushin and Ya. G. Sinai. New York-Basel 1980, 285-325. MR 83k:60016
  • 3. S. P. Humphries Cogrowth of groups and the Dedekind-Frobenius group determinant. Math. Proc. of Camb. Phil. Soc. 121 (1997), 193-217. MR 97i:20037
  • 4. H. Kesten Symmetric random walks on groups Trans. Amer. Math. Soc. 92(1959), 146-156. MR 22:253
  • 5. W. Kuich, A. Salomaa Semirings, automata, languages. Berlin; New York: Springer-Verlag, 1986. MR 87h:68093
  • 6. G. Quenell Combinatorics of free product graphs In: Geometry of the Spectrum. Contemporary Math. 173(1994), 257-281. MR 95k:05089
  • 7. W. Woess Cogrowth of groups and simple random walks. Arch. Math. 41(1983), 363-370. MR 86h:60133

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20F05, 20P05, 05C38

Retrieve articles in all journals with MSC (1991): 20F05, 20P05, 05C38

Additional Information

Dmitri Kouksov
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Keywords: Cogrowth of groups, rational series
Received by editor(s): March 13, 1997
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society