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thank you for this. I might have to refer to one of these books which discuss the action of $\mathrm{SL}_2(\mathbb{R})$ on the upper half plane. I am not very knowledgeable in linear algebra, could you tell me what some of those books are (possibly with reference to a chapter/chapters which discuss this specific classification into elliptic/parabolic/hyperbolic elements) ? Thank you very much for your continued help.
Thank you for your detailed response! Do you have a source I can use? As I am trying to utilize these facts in a paper without proving them myself. Also, for the cubic roots, is there a nice description like there is for square roots? i.e. Can I classify cubic roots similarly to how you classified the square roots?
Thank you. As a follow up, can I ask if these conjugacy classes contain infinitely many distinct elements in $\mathrm{SL}_2(\mathbb{Z})$ ? Furthermore, would it be easy to determine all the roots (not just square) of $-\mathrm{Id}_2$ inside $\mathrm{SL}_2(\mathbb{Z})$ ?
