## Some remarks on non-discrete Möbius groups.(English)Zbl 0856.30032

In this paper the author explores the consequences of the theory of Fuchsian groups for the theory of representations of a given group in $$SL(2, \mathbb{R})$$ and $$SL(2, \mathbb{C})$$. When the embedded group is discrete one can use a number of geometric arguments to determine various properties. The author observes that these embeddings can be analytically joined to an arbitrary one. He proves first the “uniqueness principle” (a sort of “principle of analytic continuation” in the discrete case and which is closely related to a theorem of Selberg and Weil. He then goes on to investigate the case of a group generated by two parabolic elements. In the case of $$SL(2, \mathbb{C})$$ he characterizes pairs of parabolic generators using geometric and specialization arguments. In the case of $$SL(2, \mathbb{R})$$ he characterizes when the group generated is discrete by the set of traces. He also indicates a number of open questions.

### MSC:

 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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