  Let $X$ be a (smooth) compact complex surface. We show that the torsion
subgroup of the biholomorphic automorphism group $\operatorname{Aut}(X)$ is
virtually nilpotent. Moreover, we study the Tits alternative of
$\operatorname{Aut}(X)$ and virtual derived length of virtually solvable
subgroups of $\operatorname{Aut}(X)$.
