Quadratic Equations in Graph Products of Groups and the Exponent of Periodicity

TL;DR

This paper explores the relationship between infinite solutions and the exponent of periodicity in quadratic equations within various finitely generated groups, extending Makanin's work from free monoids.

Contribution

It identifies structural conditions on groups that ensure unbounded exponents of periodicity in infinite solution sets of quadratic equations, and shows these conditions are preserved under graph products.

Findings

Conditions hold for all finitely generated right-angled Artin groups.

Conditions also hold for torsion-free nilpotent and hyperbolic groups.

Characterizes Baumslag-Solitar groups satisfying these conditions.

Abstract

In 1977, Makanin established the decidability of equations in free monoids. A key ingredient in his proof is the exponent of periodicity: for a word $w$, it is the largest exponent $e$ such that $w$ contains a nonempty factor of the form $p^e$. Makanin showed the following for a system of equations in free monoids: if the system has a solution with a sufficiently large exponent of periodicity, then it has infinitely many solutions. However, the converse -- whether the existence of infinitely many solutions implies the existence of solutions with arbitrarily large exponent of periodicity -- remains open. In this paper, we investigate the analogous problem for quadratic equations in finitely generated groups. We use normal forms to define the exponent of periodicity. We then identify structural conditions on groups and their normal forms that guarantee that infinite solution sets of quadratic systems have an unbounded exponent of periodicity. We prove that these conditions are preserved under graph products and, in particular, hold for all finitely generated right-angled Artin groups. In addition, we show that they also hold for finitely generated (graph products of) torsion-free nilpotent and hyperbolic groups, and we characterize the Baumslag-Solitar groups satisfying them.