Test sequences and formal solutions over hyperbolic groups

In 2006 Z. Sela and independently O. Kharlampovich and A. Myasnikov gave a solution to the Tarski problems by showing that two non-abelian free groups have the same elementary theory. Subsequently Z. Sela generalized the techniques used in his proof of the Tarski conjecture to classify all finitely generated groups elementary equivalent to a given torsion-free hyperbolic group. One important step in his analysis of the elementary theory of free and torsion-free hyperbolic groups is the Generalized Merzlyakov's Theorem. In our work we show that given a hyperbolic group $\Gamma$ and $\Gamma$-limit group $L$, there exists a larger group $Comp(L)$, namely its completion, into which $L$ embeds, and a sequence of points $(\lambda_n)$ in the variety $Hom(Comp(L),\Gamma)$ from which one can recover the structure of the group $Comp(L)$. Using such a test sequence $(\lambda_n)$ we are finally able to prove a version of the Generalized Merzlyakov's Theorem over all hyperbolic groups (possibly with torsion).