## Asian Journal of Mathematics

- Asian J. Math.
- Volume 8, Number 4 (2004), 837-860.

### Special Cycles and Automorphic Forms on Arithmetically Defined Hyperbolic 3-Manifolds

#### Abstract

An orientable hyperbolic 3-manifold is isometric to the quotient of hyperbolic 3-space H3 by a discrete torsion free subgroup G of the group Iso(H3)0 of orientation -- preserving isometries of H3. The latter group is isomorphic to the (connected) group PGL2(C), the real Lie group SL2(C) modulo its center $\pm 1$. Generally, a discrete subgroup of PGL2(C) is called a Kleinian group. The group G is said to have finite covolume if H3/G has finite volume, and is said to be cocompact if H3/G is compact. Among hyperbolic 3-manifolds, the ones originating with arithmetically defined Kleinian groups form a class of special interest. Such an arithmetically defined 3-manifold H3/G is essentially determined (up to commensurability) by an algebraic number field k with exactly one complex place, an arbitrary (but possibly empty) set of real places and a quaternion algebra Dover k which ramifies (at least) at all real places of k. These arithmetic Kleinian groups fall naturally into two classes ...

#### Article information

**Source**

Asian J. Math. Volume 8, Number 4 (2004), 837-860.

**Dates**

First available: 13 June 2005

**Permanent link to this document**

http://projecteuclid.org/euclid.ajm/1118669703

**Mathematical Reviews number (MathSciNet)**

MR2127951

**Zentralblatt MATH identifier**

1072.11029

#### Citation

Schwermer, Joachim. Special Cycles and Automorphic Forms on Arithmetically Defined Hyperbolic 3-Manifolds. Asian Journal of Mathematics 8 (2004), no. 4, 837--860. http://projecteuclid.org/euclid.ajm/1118669703.