suppose S is an infinite group of linear fractional transformation , show that the set of limit points of S is not empty . I'm studying a modular form course , and I got stuck in this question ,i tried to use the stabilizer ,or to start from SL2(Z) or a discrete group and then try to generate it , but i can't do it . Any help is appreciated.
2026-03-30 16:26:36.1774887996
Proving that The set of limit points is not empty in an infinite group of linear fractional transformation.
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I think I know the answer : since S is an infinite group then I can find an infinite sequence Vn.z then it's either bounded or not , if it's bounded then by Bolzano weistrass there exists a subsequence that converges to some z0 so Vnm.z converges to z0 . If it's not bounded then Vn.z converges to infinity and then infinity is a limit point so in both cases the set of limit points is not empty .