Growth functions for some nonautomatic Baumslag-Solitar groups

Abstract

The growth function of a group is a generating function whose coefficients a n {a_n} are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form ⟨ a , b | a − 1 b a = a 2 ⟩ \langle a,b|{a^{ - 1}}ba = {a^2}\rangle based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic. The results in this paper generalize to the groups ⟨ a , b | a − 1 b a = a m ⟩ \langle a,b|{a^{ - 1}}ba = {a^m}\rangle for all positive integers m.