Abstract
The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U (2, 1).
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Aljadeff E., Sonn J. (1998) Bounds on orders of finite subgroups of PGL n (K). J. Algebra 210(1): 352–360
Amitsur S.A. (1955) Finite subgroups of division rings. Trans. Amer. Math. Soc. 80, 361–386
Benson C.T., Grove L.C. (1970) Generators and relations for Coxeter groups. Proc. Amer. Math. Soc. 24, 545–547
Boothby W.M., Wang H.-C. (1965) On the finite subgroups of connected Lie groups. Comment. Math. Helv. 39, 281–294
Brauer R., Feit W. (1966) An analogue of Jordan’s theorem in characteristic p. Ann. of Math. 84(2): 119–131
Broué M., Malle G., Rouquier R. (1998) Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math. 500, 127–190
Coxeter H.S.M. (1974) Regular Complex Polytopes. Cambridge University Press, London
Du Val P. (1964) Homographies, Quaternions and Rotations. Oxford Mathematical Monographs. Clarendon Press, Oxford
Falbel E., Paupert J. (2004) Fundamental domains for finite subgroups in U(2) and configurations of Lagrangians. Geom. Dedicata 109, 221–238
Feit W. (1997) Finite linear groups and theorems of Minkowski and Schur. Proc. Amer. Math. Soc. 125(5): 1259–1262
Friedland S. (1997) The maximal orders of finite subgroups in GL n (Q). Proc. Amer. Math. Soc. 125(12): 3519–3526
Harris, M., Taylor, R. The Geometry and Cohomology of Some Simple Shimura Varieties. A. Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ (2001). (With an appendix by Vladimir G. Berkovich)
Herstein I.N. (1953) Finite multiplicative subgroups in division rings. Pacific J. Math. 3, 121–126
Jordan C.J. (1878) Reine Angew. Math. 84, 89–215
Kazhdan D. (1977) Some applications of the Weil representation. J. Anal. Matt. 32, 235–248
Klingler B. (2003) Sur la rigidité de certains groupes fondamentaux, l’arithméticité des réseaux hyperboliques complexes, et les “faux plans projectifs”. Invent. Math. 153(1): 105–143
Knapp A.W. (1996) Lie Groups Beyond an Introduction. Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA
Maclachlan C., Reid A.W. (2003) The Arithmetic of Hyperbolic 3-manifolds. Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York
Mumford D. (1979) An algebraic surface with K ample, (K 2)=9, p g = q = 0. Amer. J. Math. 101, 233–244
Platonov V., Rapinchuk A. (1994) Algebraic Groups and Number Theory. Pure and Applied Mathematics. Academic Press Inc., Boston, MA
Ratcliffe J.G., Tschantz S.T. (1999) On the torsion of the group O(n ,1; Z) of integral Lorentzian (n+1)×(n+1) matrices. J. Pure Appl. Algebra 136, 157–181
Reznikov A. (1995) Simpson’s theory and superrigidity of complex hyperbolic lattices. C. R. Acad. Sci. Paris Sér. I Math. 320(9): 1061–1064
Rogawski J.D. (1990) Automorphic Representations of Unitary Groups in Three Variables. Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ
Scharlau W. (1985) Quadratic and Hermitian Forms. Grundlehren Math. Wiss., vol. 270, Springer-Verlag, Berlin
Serre, J.-P. Linear Representations of Finite Groups. Springer-Verlag, New York (1977) Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, vol. 42
Shephard G.C., Todd J.A. (1954) Finite unitary reflection groups. Canadian J. Math. 6, 274–304
Stover, M. Property (FA) and lattices in SU(2, 1), Preprint.
Vignéras M.-F. (1980) Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics, vol. 800, Springer, Berlin
Weisfeiler, B. Post-classification version of Jordan’s theorem on finite linear groups. Proc. Nat. Acad. Sci. U.S.A. 81(16), (1984) Phys. Sci. 5278–5279.
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McReynolds, D.B. Finite Subgroups of Arithmetic Lattices in U (2, 1). Geom Dedicata 122, 135–144 (2006). https://doi.org/10.1007/s10711-006-9062-3
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DOI: https://doi.org/10.1007/s10711-006-9062-3