Manifolds allowing RET arithmetic.

Arithmetic of Hyperbolic Manifolds

By a hyperbolic 3-manifold we mean a complete orientable hyperbolic 3-manifold of finite volume, that is a quotient H/Γ with Γ ⊂ PSL2C a discrete subgroup of finite covolume (here briefly “a Kleinian group”). Among hyperbolic 3-manifolds, the arithmetic ones form an interesting, and in many ways more tractable, subclass. The tractability comes from the availability of arithmetic tools and invar...

Embedding arithmetic hyperbolic manifolds

We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n+ 1)-manifold or its universal mod 2 abelian cover can.

Arithmetic of hyperbolic 3-manifolds

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the “arithmetic topology”. Given 3-dimensional manifold M fibering over the circle we introduce an algebraic number field K = Q( √ d), where d > 0 is an integer number (discriminant) uniquely determined by M . The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that V o...

Hidden symmetries and arithmetic manifolds

Let M be a closed, locally symmetric Riemannian manifold of nonpositive curvature with no local torus factors; for example take M to be a hyperbolic manifold. Equivalently, M = K\G/Γ where G is a semisimple Lie group and Γ is a cocompact lattice in G. For simplicity, we will always assume that Γ is irreducible, or equivalently that M is not finitely covered by a smooth product; we will also ass...

Arithmetic Manifolds of Positive Scalar Curvature