Abstract
In this paper, we provide the complete list of all closed groups G of automorphisms of a product R of hyperbolic Riemann surfaces such that the order of any element in $$\textit{G/G}_e$$ , where $$G_e$$ is the identity component of G, is finite. In particular, if X is an analytic subvariety of R then the identity component of the stabilizer of X in $${\text {Aut}}R$$ is on this list. In its turn, it allows us to state that the identity component of the group $${\text {Aut}}X$$ must contain a group from this list.
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