Equations in virtually abelian groups: languages and growth

Abstract

This paper explores the nature of the solution sets of systems of equations in virtually abelian groups. We view this question from two angles. From a formal language perspective, we prove that the set of solutions to a system of equations forms an EDT0L language, with respect to a natural normal form. Looking at growth, we show that the growth series of the language of solutions is rational. Furthermore, considering the set of solutions as a set of tuples of group elements, we show that it has rational relative growth series with respect to any finite generating set.

EDT0L solutions to equations in group extensions

“…In this section we expand the work of Evetts and the author [19], to show that systems of equations with rational constraints in virtually abelian groups have EDT0L solution languages. We start by looking at rational sets in free abelian groups and show that systems of twisted equations in free abelian groups have EDT0L solution languages.…”

“…We can now define ET0L and EDT0L languages. We base our definitions on [19], however there are a number of equivalent definitions used elswhere.…”

“…Proof We can use the same proof that is used in [19], Proposition 3.10, using Lemma 4.1 in place of the Lemma 3.9 used in [19].…”

“…In hyperbolic groups, the addition of rational constraints was shown to preserve the fact that systems of equations have EDT0L solutions if the rational constraints were quasi-isometrically embedded [5]. We generalise the result on systems of equations in virtually abelian groups in [19] to include rational constraints.…”

“…Solutions to systems of equations in right-angled Artin groups were then shown to be EDT0L by Diekert, Jeż and Kufleitner [12]. Virtually free groups [11], hyperbolic groups [5], and virtually abelian groups [19] followed later.…”