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# Asian Journal of Mathematics

## Volume 8 (2004)

### Number 4

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

Pages: 837 – 860

DOI: http://dx.doi.org/10.4310/AJM.2004.v8.n4.a27

#### Author

#### 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 ...