Quantifier elimination for hyperbolic groups with torsion

Establish whether hyperbolic groups with torsion admit quantifier elimination down to the Boolean algebra of AE-definable sets, analogously to the quantifier elimination proved by Sela for torsion-free hyperbolic groups.

Background

Sela proved that definable sets in torsion-free hyperbolic groups can be described via Boolean combinations of AE-formulas, yielding strong model-theoretic control and consequences such as homogeneity characterizations. In the presence of torsion, such a quantifier elimination is not known and the authors present phenomena (e.g., non-EAE-homogeneous Coxeter groups) that suggest richer complexity.

Determining whether an analogous quantifier elimination holds for hyperbolic groups with torsion would significantly advance the model theory of these groups and impact questions about homogeneity and elementary embeddings.