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Some properties of Poincaré series of dimension \(-2\). (Einige Eigenschaften der Poincaréschen Reihen der Dimension \(-2\).) (German) Zbl 0264.30024

Let \(G\) be a Fuchsian group acting on the unit disc \(E\) of the complex plane. Using an argument, which is essentially due to M. Tsuji [Potential theory in modern function theory. Tokyo: Maruzen (1959; Zbl 0087.28401)], the reviewer [J. Math. Mech. 18, 629–644 (1969; Zbl 0184.11301)] showed that if \(\sum_{T\in G} (1-| Tz|^2)\) converges in \(E\), then it is a bounded function of \(z\) on \(E\), provided \(G\) has no elliptic elements. The author observes that the result and (essentially) the same proof remain valid even if \(G\) contains elliptic elements. Then he points out that this observation, together with the fact that a fundamental region for a finitely generated \(G\) of first kind has finite hyperbolic area, leads to a simple proof of the known result that \(\sum_{T\in G} (1-| Tz|^2)\) diverges for such groups \(G\).

MSC:

11F12 Automorphic forms, one variable
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11F06 Structure of modular groups and generalizations; arithmetic groups
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References:

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