Languages, groups and equations

The survey provides an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages. We begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to systems of equations in free groups in terms of reduced words are expressible as EDT0L languages. We provide a sketch of their algorithm, and describe how the free group results extend to hyperbolic groups. The characterisation of solutions as EDT0L languages is very robust, and many group constructions preserve this, as shown by Levine. The most recent progress in the area has been made for groups without negative curvature, such as virtually abelian, the integral Heisenberg group, or the soluble Baumslag-Solitar groups, where the approaches to describing the solutions are different from the negative curvature groups. In virtually abelian groups the solutions sets are in fact rational, and one can obtain them as $m$-regular sets. In the Heisenberg group producing the solutions to a single equation reduces to understanding the solutions to quadratic Diophantine equations and uses number theoretic techniques. In the Baumslag-Solitar groups the methods are combinatorial, and focus on the interplay of normal forms to solve particular classes of equations. In conclusion, EDT0L languages give an effective and simple combinatorial characterisation of sets of seemingly high complexity in many important classes of groups.Comment: 26 page

Solving equations in class 2 nilpotent groups

We construct an algorithm to decide if in a class $2$ nilpotent group an equation $\omega = 1$ that contains a variable $X$ such that the exponent sum of $X$ within $\omega$ is non-zero admits a solution. Besides the existence of such an $X$ in $\omega$, there are no restrictions on any other variables. We do this by associating the equation to a system of integer equations and congruences equivalent to it, and give an algorithm to solve this system. We also construct an algorithm to decide if any equation in a group that is virtually the Heisenberg group admits a solution.Comment: 21 pages Fixed error in main theore

Solutions sets to systems of equations in hyperbolic groups are EDT0L in PSPACE

© Laura Ciobanu and Murray Elder; licensed under Creative Commons License CC-BY We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in NSPACE(n2 log n) for the torsion-free case and NSPACE(n4 log n) for the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Jeż, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and an intricate language-theoretic analysis. The present work gives an essentially optimal formal language description for all solutions in all hyperbolic groups, and an explicit and surprising low space complexity to compute them