Definable sets in a hyperbolic group

TL;DR

This paper characterizes definable sets in free non-abelian and torsion-free hyperbolic groups, answering longstanding questions and revealing that proper non-cyclic subgroups are not definable, with implications for subgroup size classifications.

Contribution

It provides a description of definable sets in free and hyperbolic groups, resolving Malcev's question and confirming that certain subgroups are not definable.

Findings

Proper non-cyclic subgroups are not definable in these groups.

Definable subsets in free groups are either negligible or co-negligible.

The work answers Malcev's question for free groups.

Abstract

We give a description of definable sets $P=(p_1,..., p_m)$ in a free non-abelian group $F$ and in a torsion-free non-elementary hyperbolic group $G$ that follows from our work on the Tarski problems. This answers Malcev's question for $F$. As a corollary we show that proper non-cyclic subgroups of $F$ and $G$ are not definable and prove Bestvina and Feighn's result that definable subsets $P=(p)$ in a free group are either negligible or co-negligible in their terminology.