Effective equation solving, constraints and growth in virtually abelian groups

TL;DR

This paper demonstrates that solving equations with various constraints in virtually abelian groups is decidable and provides methods to effectively compute solutions and the group's growth series, extending known theoretical results.

Contribution

It introduces effective methods to solve constrained equations and compute growth series in virtually abelian groups, translating constraints into rational sets.

Findings

Solutions with added constraints can be effectively produced.

Constraints can be translated into rational sets, reducing the problem.

Growth series of virtually abelian groups is effectively computable.

Abstract

In this paper we study the satisfiability and solutions of group equations when combinatorial, algebraic and language-theoretic constraints are imposed on the solutions. We show that the solutions to equations with length, lexicographic order, abelianisation or context-free constraints added, can be effectively produced in finitely generated virtually abelian groups. Crucially, we translate each of the constraints above into a rational set in an effective way, and so reduce each problem to solving equations with rational constraints, which is decidable and well understood in virtually abelian groups. A byproduct of our results is that the growth series of a virtually abelian group, with respect to any generating set and any weight, is effectively computable. This series is known to be rational by a result of Benson, but his proof is non-constructive.