Definable sets in a hyperbolic group
  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. 
Olga Kharlampovich, Alexei Myasnikov
math.GR