Abstract
Abstract For 𝓞 an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE2(𝓞), the projective elementary subgroup of PSL2(𝓞). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE2(𝓞), show that it has infinite index and is its own normalizer in PSL2(𝓞), and split PSL2(𝓞) into a free product with amalgamation that has PE2(𝓞) as one of its factors.
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