Keywords:-

Keywords: .

Article Content:-

Abstract

Selected topic in the cohomology of non-arithmetical groups algebras and grouppal structures are chosen and
reviewed. The basics formalisms are recalled. The comparisons between the different approaches are proposed,
from which new open questions arise; in particular, the application of the method of satellites to the study of non-
arithmetical grouppal structures is evidentiated to request the definition of new structures in the calculation of the
irreducible characters.

References:-

References

M. Kuga, W. Parry, Ch. Sah, Group Cohomology and Hecke Operators, in: Hano J., Morimoto A., Murakami S., Okamoto

K., Ozeki H. (eds) Manifolds and Lie Groups. Progress in Mathematics, vol 14., pp. 223-266, Birkhaeuser, Boston, MA

(1981).

Y.H. Rhie, G. Whaples, Hecke operators in cohomology of groups, J. Math. Soc. Japan 22, 431 (1970).

T. Yoshida, On G-functors (II): Hecke operators and G-functors, J. Math. Soc. Japan 35, 179 (1983).

H. Movasati, K.M. Shokri, Automorphic forms for triangle groups: Integrality properties, Journal of Number Theory 145,

(2014).

Problems on Mapping Class Groups and Related Topics, Ed.: Benson Farb, Proceedings of Symposia in Pure Mathematics

Volume: 74; AMS (2006).

D. Margalit, problems, questions and conjectures about mapping calss groups, eprint [arXiv:1806.08773] (2018).

R. J. Lawrence, Homological representations of the Hecke algebra. Comm. Math. Phys., 135, 141 (1990).

S. Morita, Characteristic classes of surface bundles, Invent. Math. 90, 551 (1987).

D. Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr.

Math., vol. 36, pp. 271-328, Birkhaeuser Boston, Boston, MA (1983).

E. Formanek, C. Procesi, Mumford’s Conjecture for the General Linear Group, Advances in Mathematics 19, 292 (1976).

V.H. Moll, S. Robins, K. Soodhalter, The action of Hecke operators on hypergeometric functions, eprint [arXiv:1005.2946]

(2010).

C.F. Doran, T. Gannon, H. Movasati, K.M. Shokri, Automorphic forms for triangle groups, Commun. Num. Theor Phys.

, 689 (2013).

S. Akiyama, A note on Hecke’s absolute invariants, J. Ramanujan Math. Soc. 7, 65 (1992).

J. Raleigh, On the Fourier coefficients of automorphic forms of triangle groups, Acta Arith. 8, 107 (1962).

S. Akiyama, On the Fourier coefficients of automorphic forms of triangle groups,Kobe J. Math. 5, 123 (1988).

A.J. Scholl, Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences, Invent. Math. 79, 49 (1985).

A.O.L. Atkin, H.P.F. Swinnerton-Dyer, Modular forms on noncongruence subgroups. In: Proc.Symp.Pure.Math. (AMS,

Providence), ed. by T.S. Motzkin, pp. 1-26 (1971).

I.I. Bouw, M. Moeller, Differential equations associated with nonarithmetic Fuchsian groups, Journal of the London

Mathematical Society, 81, 65 (2010).

R.C. Orellana, Weights of Markov Traces on Hecke Algebras, arXiv:math/9806163 [math.CO].

A. de Rham, Accessing the cohomology of discrete groups above their virtual cohomological dimension, eprint

[arXiv:1112.4262 [math.KT]].

G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology and its Applications 56, 293 (1994).

S. Lambropoulou, M. Geck, Markov traces and knot invariants related to Iwahori-Hecke algebras of type B, Journal fuer

die reine und angewandte Mathematik (Crelles Journal) 482, 191 (1997).

M. Geck, Trace functions on Iwahori-Hecke algebras, Banach Center Publications 42, 87 (1998).

M. Geck, G. Pfeiffer, On the Irreducible Characters of Hecke Algebras, Advances in Mathematics 102, 79 (1993).

K. Imai and A. Terras, The Fourier Expansion of Eisenstein Series for GL(3, Z), Transactions of the American Mathematical

Society, 273, 679 (1982).

F. Grunewald, V. Platonov, Non-arithmetic polycyclic groups [Groupes polycycliques non arithm ́etiques] Comptes Rendus

de l’Acad ́emie des Sciences-Series I-Mathematics 326, 1359 (1998).

K. Dekimpe, P. Igodt, Polynomial Structures on Polycyclic Groups, Transactions of the American Mathematical Society,

, 3597 (1997).

H. Nagoshi, Distribution of Hecke Eigenvalues, Proceedings of the American Mathematical Society 134, 3097 (2006).

J. B. Conrey, W. Duke, D.W. Farmer, The distribution of the eigenvalues of Hecke operators, Acta Arithmetica 78, 405

(1997).

G. Steil, Eigenvalues of the Laplacian and of the Hecke operators for PSL(2, Z), Hamburg University Report number:

DESY-94-028 (1994).

H. Avelin, Computations of Eisenstein series on Fuchsian groups, Mathematics of Computation 77, 1779 (2008).

M. Schuett, Hecke eigenforms with rational coefficients and complex multiplication, Corpus ID: 14682518 (2005).

A. Doikou, Asymmetric twin representation: The Transfer matrix symmetry, SIGMA 3, 009 (2007).

Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications I, Springer-Verlag New York Inc., New York,

USA (1985).

A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verlag New York Inc., New York, USA

(1985).

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Lecian, O. (2022). Selected topics in Cohomology of non-arithmetical groups (algebras) and open questions. International Journal Of Mathematics And Computer Research, 10(12), 3002 -3014. https://doi.org/10.47191/ijmcr/v10i12.03