## Experimental Mathematics

### Growth functions and automatic groups

#### Abstract

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

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

Dates
First available in Project Euclid: 13 March 2003

https://projecteuclid.org/euclid.em/1047565448

Mathematical Reviews number (MathSciNet)
MR1437220

Zentralblatt MATH identifier
0892.20022

Subjects
Primary: 20F32

#### Citation

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