×

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

MSC:

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

Citations:

Zbl 0605.22008
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[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
[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
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.