The outer-automorphism group of Z^2 is confirmed to be SL(2,Z). The discussion clarifies that the automorphism group, Aut(Z^2), is equivalent to GL(2,Z), while the inner automorphism group, Inn(Z^2), is trivial, resulting in Out(Z^2) being equal to GL(2,Z). The confusion arose from incorrectly applying the Dehn-Nielsen theorem, which is applicable only to hyperbolic surfaces, to the torus, a parabolic surface. This distinction is crucial for understanding the structure of these groups.
PREREQUISITESThis discussion is beneficial for mathematicians, particularly those specializing in algebra and topology, as well as students seeking to deepen their understanding of group theory and its applications in geometry.