Experimental Mathematics

Growth functions and automatic groups

David B. A. Epstein, Anthony R. Iano-Fletcher, and Uri Zwick


In this paper we study growth functions of automatic and hyperbolic groups. In addition to standard growth functions, we also want to count the number of finite graphs isomorphic to a given finite graph in the ball of radius $n$ around the identity element in the Cayley graph. This topic was introduced to us by K. Saito [1991]. We report on fast methods to compute the growth function once we know the automatic structure. We prove that for a geodesic automatic structure, the growth function for any fixed finite connected graph is a rational function. For a word-hyperbolic group, we show that one can choose the denominator of the rational function independently of the finite graph.

Article information

Experiment. Math., Volume 5, Issue 4 (1996), 297-315.

First available in Project Euclid: 13 March 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F32


Epstein, David B. A.; Iano-Fletcher, Anthony R.; Zwick, Uri. Growth functions and automatic groups. Experiment. Math. 5 (1996), no. 4, 297--315. https://projecteuclid.org/euclid.em/1047565448

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