Hypergeometric Groups: arithmetic or thin?

The hypergeometric groups are the subgroups of $GL_n(C)$ generated by the companion matrices of two monic coprime polynomials of degree $n$ and when the hypergeometric groups are infinite their Zariski closures (inside $GL_n(C)$) are either the symplectic or the orthogonal groups. In this talk we present the progress on the question to determine the arithmetic or thin hypergeometric groups.