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Zbl 0649.22007
Gromov, M.; Piatetski-Shapiro, Ilya I.
Non-arithmetic groups in Lobachevsky spaces. (English)
[J] Publ. Math., Inst. Hautes Étud. Sci. 66, 93-103 (1988). ISSN 0073-8301; ISSN 1618-1913/e

A result of Margulis says that every lattice in a simple Lie group G with rank$\sb{{\bbfR}}G>2$ is arithmetic. Up to local isomorphism it remains to consider the following non-compact groups (groups with rank$\sb{{\bbfR}}=1):$ $O(n,1)$, $U(n,1)$, and their quaternion and Cayley analogues. Non-arithmetic lattices in $SU(2,1)$ and $SU(3,1)$ were constructed by {\it 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. \par 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 $V\sp+\sb i$ with boundary of dimension n of the corresponding hyperbolic manifolds $V\sb i$ along the (n-1)-dimensional boundary $\partial V\sb i\sp+$ (which is assumed to be totally geodesic in $V\sb i)$ by means of an isometry $\partial V\sb 1\sp+{\tilde \to}\partial V\sb 2\sp+$ 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 $\Gamma\sp+\sb i$ of $V\sp+\sb i$ 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 $\Gamma\sb i$ are commensurable. In turn, one obtains a non-arithmetic lattice $\Gamma$ by starting with two non-commensurable groups $\Gamma\sb 1$ and $\Gamma\sb 2$.
[J.Schwermer]

MSC 2000:: *22E40 Discrete subgroups of Lie groups
22E46 Semi-simple Lie groups and their representations

Keywords: non-arithmetic lattices; projective orthogonal group; arithmetic subgroups; hyperbolic manifolds; universal covering; fundamental group; commensurability

Citations: Zbl 0605.22008

Cited in: Zbl 1122.14039 Zbl 1113.57007 Zbl 1033.57012 Zbl 0995.53024 Zbl 0876.22015

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