Core Concepts
In virtually abelian groups, it is effectively decidable whether a finite system of group equations has solutions satisfying various constraints, including linear length, abelianization, context-free, and lexicographic order constraints. The key is that these seemingly non-rational constraints can be effectively translated into rational sets in virtually abelian groups.
Abstract
The paper studies the satisfiability and solutions of group equations when combinatorial, algebraic, and language-theoretic constraints are imposed on the solutions in virtually abelian groups. The main results are:
Theorem 1.1: In any finitely generated virtually abelian group, it is effectively decidable whether a finite system of equations with the following kinds of constraints has solutions:
Linear length constraints (with respect to any weighted word metric)
Abelianization constraints
Context-free constraints
Lexicographic order constraints
Theorem 1.2: In any finitely generated virtually abelian group, there is an effective way to construct a rational set from the above constraints.
Corollary 1.3: The set of solutions to a finite system of equations with the above constraints can be expressed as an EDT0L language in at least two different ways.
Proposition 1.4: The weighted growth series (with respect to any finite generating set) of a finitely generated virtually abelian group can be computed explicitly. This was previously known to be rational, but the proof was non-constructive.
The paper combines group theory, theoretical computer science, and combinatorics to study these equation solving problems in virtually abelian groups. The key insight is that seemingly non-rational constraints like length, abelianization, and context-free constraints turn out to be rational in this setting, allowing for decidability results.