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Toward a Bianchi-Quaternion Dictionary

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Senia Sheydvasser (CUNY)

Abstract: Given a Bianchi group SL(2, o_K), it is well-known how to compute geometric information about the quotient manifold H^3/SL(2, o_K) from the arithmetic of the ring of integers o_K; this has important consequences both from the perspective of geometry and number theory. In this talk, we will discuss a modern attempt to create an analogous theory for hyperbolic 4-manifolds using orders of quaternion algebras.