Equations in groups that are virtually direct products

Abstract

In this note, we show that the satisfiability of equations and inequations with recognisable constraints is decidable in groups that are virtually direct products of finitely many hyperbolic groups.Dedicated to Charles Sims, who introduced the first author to equations in groups.

Acylindrical hyperbolicity for Artin groups of dimension 2

“…Proof It is known that the quotient Āab of A ab by its centre is virtually isomorphic to the free group F m , for some m ≥ 2 [6,15]. In particular, Āab is acylindrically hyperbolic.…”

“…This also holds if A is reducible, as 2-dimensional reducible Artin groups are direct products of free groups, hence have trivial centres. If A is irreducible and has rank 2 then it is a dihedral Artin group with coefficient at least 3, and A /Z (A ) is virtually a free group [6,15], hence acylindrically hyperbolic.…”

EDT0L solutions to equations in group extensions

“…In this section, we show that solution languages to systems of equations in groups that are virtually direct products of hyperbolic groups are EDT0L. We adapt the method that Ciobanu, Holt and Rees use to show that the satisfiability of systems of equations in these groups is decidable [7]. For an introduction to hyperbolic groups, we refer the reader to [22], Chapter 6.…”

“…The proof of Theorem A( 4) is based on Ciobanu, Holt and Rees' proof of the fact the satisfiability of systems of equations in these groups is decidable [7], in a work that also looks at recognisable constraints. We show that the addition of recognisable constraints to any system of equations preserves the property of having an EDT0L solution language, and use this to show that the class of groups where systems of equations have EDT0L solutions is closed under passing to finite index subgroups.…”

“…Group equations have seen significant interest as to the decidability of the satisfiability of equations in specific classes of groups, since Makanin showed in the 1980s that the satisfiability of systems in free groups was decidable ( [25], [26], [27]). The satisfiability of systems of equations has been shown to be decidable or undecidable in a wide variety of other classes of groups ( [31], [30], [9], [13], [7], [24]). Describing the set of solutions in any meaningful way has often proved difficult.…”