Rational sets in virtually abelian groups: Languages and growth

  • Laura Ciobanu

    Heriot-Watt University, Edinburgh, UK
  • Alex Evetts

    University of Manchester, Manchester, UK; The Heilbronn Institute for Mathematical Research, Bristol, UK
Rational sets in virtually abelian groups: Languages and growth cover
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Abstract

In this paper, we generalise and unify the results and methods used by Benson, Liardet, Evetts, and Evetts &Levine to show that rational sets in a virtually abelian group have rational (relative) growth series with respect to any generating set for . We prove equivalences between the structures used in the literature and establish the rationality of important classes of sets in : definable sets, algebraic sets, conjugacy representatives, and coset representatives (of any fixed subgroup), among others. Furthermore, we show that any rational set, when written as words over the generating set of , has several EDT0L representations.

Cite this article

Laura Ciobanu, Alex Evetts, Rational sets in virtually abelian groups: Languages and growth. Enseign. Math. 71 (2025), no. 3/4, pp. 231–260

DOI 10.4171/LEM/1074