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Number Theory Seminar

Low-lying fundamental geodesics in an arithmetic hyperbolic 3-manifold

Katie McKeon (Rutgers)

Location:  Hill 525
Date & time: Tuesday, 06 February 2018 at 2:00PM - 3:00PM

We’ll examine closed geodesics in the quotient of hyperbolic three space by the discrete group of isometries SL(2,Z[i]). There is a correspondence between closed geodesics in the manifold, the complex continued fractions originally studied by Hurwitz, and binary quadratic forms over the Gaussian integers. According to this correspondence, a geodesic is called fundamen- tal if the associated binary quadratic form is. Using techniques from sieve theory, symbolic dynamics, and the theory of expander graphs, we show the existence of a compact set in the manifold containing infinitely many fundamental geodesics.