$T$-Jordan property for automorphism groups of compact complex surfaces

A group $G$ is called $T$-Jordan if there is a constant $J(G)$ such that every torsion subgroup of $G$ contains a abelian subgroup of index $\le J$. In a 1976 paper, Dong Hoon Lee showed that a connected Lie group is $T$-Jordan. Similarly to the question of when automorphism groups of varieties are Jordan, one can consider the question when they are $T$-Jordan. Following Jia Jia's paper we will discuss the $T$-Jordan property in the case of compact complex surfaces.