# Non-arithmetic lattices.

##### September 14, 2006 mathematics

Margulis’ amazing arithmeticity theorem says that irreducible lattices in Lie groups of high () rank are arithmetic. But has rank 1, so a question is how to produce non-arithmetic lattices. For , 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 ’s in 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.