# Non-arithmetic lattices.

##### 2006-09-14 16:56:03 +0000 mathematics

MR0932135

Vinberg, …

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 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.