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The Eichler cohomology of a Kleinian group. (English) Zbl 0213.10002
In this paper the Eichler cohomology is developed for all finitely generated Kleinian groups that have a finite-faced polyhedral fundamental domain.
Theorem 1.
\[ C_i^0(\Gamma, -r-2,\bar v) \oplus I_i^+ (\Gamma, r, v)/\mathcal P_r \cong H(\Gamma, r, v). \]
Theorem 2.
\[ C_i^0(\Gamma, -r-2,\bar v) \oplus I_i^0 (\Gamma, r, v)/\mathcal P_r \cong P_iH(\Gamma, r, v). \]
In these theorems \(\Gamma\) is a Kleinian group, \(\Omega_i\) is a certain \(\Gamma\)-invariant union of components of the discontinuity set \(\Omega\) of \(\Gamma\), \(r>0\) is an integer, \(v\) is a complex character of \(\Gamma\), \(H(\Gamma, r, v)\) is the first cohomology group based on the usual action of \(\Gamma\) on \(\mathcal P_r\) (the coefficient module of polynomials of degree \(\le r)\), \(I_i^+\) is a certain space of automorphic integrals of degree \(r\) and character \(v\), and \(C_i^0\) is the space of cusp forms of “complementary degree” \(-r-2\) and conjugate character \(v\).
In the second theorem \(P_iH(\Gamma,r,v)\) is the “parabolic” cohomology group determined by the cusps of \(\Gamma\) associated with \(\Omega_i\). In Section 9 these theorems are extended to larger unions of components of \(\Omega\).
These results are similar to those obtained by L. V. Ahlfors [Acta Math. 122, 1–17 (1969; Zbl 0193.04002)], L. Bers [J. Anal. Math. 18, 23–41 (1967; Zbl 0146.31601)], and I. Kra [Ann. Math. (2) 89, 533–556 (1969; Zbl 0193.04003); 90, 576–590 (1969; Zbl 0172.10602)]. In particular, the two Bers’ area theorems are obtained.
By specialization of \(\Omega_i\), one also gets previously known results on Fuchsian groups of M. Eichler [Math. Z. 67, 267–298 (1957; Zbl 0080.06003)], R. C. Gunning [Trans. Am. Math. Soc. 100, 44–62 (1961; Zbl 0142.05302)], and S. Y. Husseini and M. I. Knopp (to be published in) [Ill. J. Math. 15, 565–577 (1971; Zbl 0224.10024)].
The method used depends heavily on the construction of automorphic integrals by means of Poincaré-type series, as originally envisaged by Eichler in the reference above (cf. also the author [J. Res. Natl. Bur. Stand., Sect. B 73, 153–161 (1969; Zbl 0185.33501)]). The Riemann-Roch theorem is also used in an essential way to reduce the singularities of the integrals.
Reviewer: Joseph Lehner

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
11F75 Cohomology of arithmetic groups
Full Text: DOI EuDML
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