Non-arithmetic lattices.

 September 14, 2006 mathematics

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Margulis’ amazing arithmeticity theorem says that irreducible lattices in Lie groups of high ( >2 ) rank are arithmetic. But {\rm SO}(n,1) has rank 1, so a question is how to produce non-arithmetic lattices. For {\rm SO}(3,1) , there are non-arithmetic lattices coming from hyperbolic knot complements.

G–P-S [1] produces higher dimensional examples by taking two hyperbolic (arithmetic) manifolds, cutting along totally geodesic hypersurfaces, and gluing. Are there are examples of non-arithmetic hyperbolic manifolds without any totally geodesics hypersurfaces?

There are complements of T^2 ’s in S^4 which are hyperbolic, and maybe these would provide some examples.

[1] M. Gromov, I. Piatetski-Shapiro, Nonarithmetic groups in lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. (1988) 93–103.