## 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:

 2.2e+41 Discrete subgroups of Lie groups 2.2e+47 Semisimple Lie groups and their representations

Zbl 0605.22008
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### References:

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