Automorphism groups of compact complex surfaces: T-Jordan property, Tits alternative and solvability

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 $\operatorname{Aut}(X)$.