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
