Non-arithmetic groups in Lobachevsky spaces. (English) Zbl 0649.22007

A result of Margulis says that every lattice in a simple Lie group G with rank\(_{{\mathbb{R}}}G>2\) is arithmetic. Up to local isomorphism it remains to consider the following non-compact groups (groups with rank\(_{{\mathbb{R}}}=1):\) \(O(n,1)\), \(U(n,1)\), and their quaternion and Cayley analogues. Non-arithmetic lattices in \(SU(2,1)\) and \(SU(3,1)\) were constructed by G. Mostov using reflections in complex hyperplanes [cf. Elie Cartan et les mathématiques d’aujourd’hui, Astérisque, No.Hors. Sér. 1985, 289-309 (1985; Zbl 0605.22008)]. In the other case of the hyperbolic space examples of non-arithmetic lattices (for \(n=3,4,5)\) were found by Makarov, Nikulin and Vinberg.
The paper under review provides a general construction of non-arithmetic lattices (cocompact and non-cocompact) in the projective orthogonal group \(PO(n,1)=O(n,1)/(\pm 1)\) for all \(n=2,3,...\). By taking two torsion free arithmetic subgroups of \(PO(n,1)\) and gluing together two submanifolds \(V^+_ i\) with boundary of dimension n of the corresponding hyperbolic manifolds \(V_ i\) along the (n-1)-dimensional boundary \(\partial V_ i^+\) (which is assumed to be totally geodesic in \(V_ i)\) by means of an isometry \(\partial V_ 1^+{\tilde \to}\partial V_ 2^+\) the authors produce a hybrid manifold V. The universal covering of V turns out to be the hyperbolic space and the fundamental group of V is a lattice in the isometry group PO(n,1) of the hyperbolic space. In the relevant cases the fundamental group \(\Gamma^+_ i\) of \(V^+_ i\) is Zariski dense in \(PO(n,1)\circ\). This implies the following commensurability property: If the group \(\Gamma\) is arithmetic then the groups \(\Gamma\) and \(\Gamma_ i\) are commensurable. In turn, one obtains a non-arithmetic lattice \(\Gamma\) by starting with two non-commensurable groups \(\Gamma_ 1\) and \(\Gamma_ 2\).
Reviewer: J.Schwermer


22E40 Discrete subgroups of Lie groups
22E46 Semisimple Lie groups and their representations


Zbl 0605.22008
Full Text: DOI Numdam EuDML


[1] A. Borel,Introduction aux groupes arithmétiques, Paris, Hermann (1969). · Zbl 0186.33202
[2] A. Borel,Linear algebraic groups, New York, Benjamin (1969). · Zbl 0206.49801
[3] M. Gromov, Hyperbolic manifolds according to Thurston and Jorgensen,Lecture Notes in Mathematics,842, Springer-Verlag (1981), pp. 40–54. · Zbl 0452.51017 · doi:10.1007/BFb0089927
[4] M. Gromov,Rigid transformation groups. To appear. · Zbl 0652.53023
[5] V. S. Makarov, On a certain class of discrete Lobachevsky space groups with infinite fundamental domain of finite measure,Dokl. Ak. Nauk. U.S.S.R.,167 (1966), pp. 30–33.
[6] G. Mostow, Discrete subgroups of Lie groups, inElie Cartan et les Mathématiques d’aujourd’hui, Astérisque, numéro hors série (1985), 289–309.
[7] V. V. Nikulin,Discrete reflection groups in Lobachevsky space and algebraic surfaces, Preprint. · Zbl 0671.22006
[8] M. S. Raghunathan,Discrete subgroups of Lie groups, Springer (1972). · Zbl 0254.22005
[9] E. B. Vinberg, Hyperbolic reflection groups,Usp. Math. Nauk.,40 (1985), pp. 29–66. · Zbl 0579.51015
[10] R. Zimmer,Ergodic theory and semisimple groups, Birkhauser, Boston (1984). · Zbl 0571.58015
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