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.

翻译：本文研究在解的组合、代数及语言理论约束下群方程的满足性与求解问题。我们证明，在有限生成的几乎阿贝尔群中，当施加长度、字典序、阿贝尔化或无上下文约束时，方程的解可被有效生成。关键在于，我们将上述每种约束以有效方式转化为有理集，从而将每个问题归约为具有有理约束的方程求解——这类问题在几乎阿贝尔群中是可判定的且已被充分理解。作为本文结果的副产品，几乎阿贝尔群关于任意生成集和任意权重的增长级数是可有效计算的。该级数由Benson的结果已证明为有理函数，但其证明是非构造性的。