Under natural extension hypotheses, discreteness of fixed point sets forces finiteness for automorphisms of bounded domains in C^n, plus uniform bounds for compact subgroups.
David Minda,Fixed points of analytic self-mappings of Riemann surfaces, Manuscripta Math.27(1979), no
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Finiteness of the fixed point sets of automorphisms
Under natural extension hypotheses, discreteness of fixed point sets forces finiteness for automorphisms of bounded domains in C^n, plus uniform bounds for compact subgroups.