Hyperbolic Manifolds from Arithmetic Pieces

Let S be a finite set of invertible 2-by-2 matrices with algebraic entries. Is there an algorithm to determine a presentation for <S>? We shall pose this question, and suggest a possible avenue to answering it. Note that every finite-volume hyperbolic 3-manifold arises as above. Beyond this, we will survey many algebraic and arithmetic aspects of hyperbolic manifolds. We will provide some conditions which guarantee that the fundamental group of a hyperbolic manifold contains an integral element, and show that in this case, the manifold has a finite cover which has an “arithmetic piece”.