Generators for exact representations of 3-manifold groups

Question

Does anyone have a list of matrix generators for a bunch of hyperbolic 3-manifold groups? I am testing an algorithm and am looking for a collection of test cases. I am looking for exact values of matrix elements as algebraic numbers, and not approximate values.

Answer 1

Grant Lakeland was kind enough to compute exact $PSL(2,\mathbb{C})$ representations for tetrahedral groups. The methods and formulas are written up on his webpage here.

Maclachlan and Reid give complete list of the arithmetic and non-arithmetic tetrahedral groups in $\S$13.1 and $\S$13.2 of their book, "The arithmetic of hyperbolic 3-manifolds."

As a warning if one should be careful about observing general phenomena from these groups. They are special a lot of ways.

However, if you want more that manifolds that not just covers of tetrahedral orbifolds, there are a few options. In the rest of section 13 of Maclachlan and Reid, (especially 'the arithmetic zoo'), they give information on the invariant trace fields of a number of manifolds. From this exact value, one can often guess and verify the algebraic numbers (most of the time these will be algebraic integers) for the entries of the generators.

Also, for general 3-manifold groups you probably want to use snap or you can use SAGE's snappy package (documentation is here) to get exact representation. Again, one has to extrapolate information from the trace field to get the entries of the generators express as algebraic numbers.