I am currently reading through Katok on fuchsian groups chapter 2.1. There the author introduces a classification of PSL(2, R) by the trace of the tranformation. It then says that hyperbolas are in one case and ellipses in the other case invariant under these maps. I don't see why this is the case and I also don't see the link between the shape of the matrix and the trace of the transformation. Unfortunately I haven't found any online source dealing with this. Could somebody please elaborate on that ? Thanks in advance.
2026-03-27 02:35:50.1774578950
What is the intuition behind the classification of möbius transformation in hyperbolic, elliptic and parabolic transformations?
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