20.91 (2022)

A group $G$ is said to be stable if for any first-order formula $\phi(\bar{x}, \bar{y})$ (in the first-order language of groups $\{\cdot, {}^{-1}, 1\}$) there exists a natural number $n$ such that whenever there exist sequences of tuples of $G$, $(a_i)_{i < m}$, $(b_i)_{i < m}$ with $G \models \phi(a_i, b_j)$ if and only if $i < j$, then $m \leqslant n$. Sela proved that all torsion-free hyperbolic groups are stable.
$\qquad$ a) Is there a non-stable hyperbolic group?
$\qquad$ b) Is there a non-stable virtually free group?
$\qquad$ c) Is $\text{SL}_2(\mathbb{Z})$ non-stable?