×

zbMATH — the first resource for mathematics

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

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
11F75 Cohomology of arithmetic groups
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Ahlfors, L. V.: Eichler integrals and Bers’ area theorem. Michigan Math. J.15, 257–263 (1968). · Zbl 0193.04001
[2] —- The structure of a finitely generated Kleinian group. Acta Math.122, 1–17 (1969). · Zbl 0193.04002
[3] Bers, L.: Inequalities for finitely generated Kleinian groups. J. Analyse Math.18, 23–41 (1967). · Zbl 0146.31601
[4] Eichler, M.: Eine Verallgemeinerung der Abelschen Integrale. Math. Zeitschrift67, 267–298 (1957). · Zbl 0080.06003
[5] —- Grenzkreisgruppen und kettenbruchartige Algorithmen. Acta Arith.11, 169–180 (1965). · Zbl 0148.32503
[6] Greenberg, L.: Fundamental polyhedra for Kleinian groups. Ann. Math.84, 433–441 (1966). · Zbl 0161.27405
[7] Kra, I.: On cohomology of Kleinian groups. Ann. Math.89, 533–556 (1969). · Zbl 0193.04003
[8] —- On cohomology of Kleinian groups: II. Ann. Math.90, 575–589 (1969). · Zbl 0182.11302
[9] Kubota, T.: Uber diskontinuierliche Gruppen Picardschen Typus und zugehörige Eisensteinsche Reihen. Nagoya Math. J.32, 259–271 (1968). · Zbl 0159.31303
[10] Petersson, H.: Automorphe Formen als metrische Invarianten. I. Math. Nachr.1, 158–212 (1948). · Zbl 0031.12503
[11] —- Über automorphe Formen mit Singularitäten im Diskontinuitätsgebiet. Math. Ann.129, 370–390 (1955). · Zbl 0068.29304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.