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

## TL;DR

This paper proves that the set of solutions to systems of equations in hyperbolic groups can be described as an EDT0L language and computed within low space complexity, combining geometric, algebraic, and language-theoretic methods.

## Contribution

It establishes that solution sets in hyperbolic groups are EDT0L languages and provides efficient space complexity algorithms for their computation.

## Key findings

- Solution sets are EDT0L languages in hyperbolic groups.
- Solution sets can be computed in NSPACE(n^2 log n) or NSPACE(n^4 log n) depending on torsion.
- The work combines geometric group theory with formal language and complexity theory.

## Abstract

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 $\mathsf{NSPACE}(n^2\log n)$ for the torsion-free case and $\mathsf{NSPACE}(n^4\log n)$ in 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, Je\.z, Diekert and others on $\mathsf{PSPACE}$ 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.