Kyler Siegel Kyler Siegel

Email: ksiegel@math.stanford.edu, kylersiegel@gmail.com
Office: 380-S, 450 Serra Mall, Stanford CA, 94305


ABOUT ME


TEACHING


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RECENT MATH PAPERS


OLDER PAPERS

  • Applying Poincare's polyhedron theorem to groups of hyperbolic isometries.

      Abstract: We present a computer algorithm which confirms that an approximately discete subgroup of PSL(2,C) is in fact discrete. The algorithm proceeds by constructing the Dirichlet domain of the subgroup in H3 , and then checks that the hypotheses of Poincares Theorem for Fundametal Polyhedra are satisfied. In order to check that the hypotheses are exactly satisfied, we rely on group theoretical properties resulting from certain geometric conditions of the Dirichlet domain. We begin in Section 1 with a review of relevant background material. In Section 2 we provide a formal statement of the problem, including the restrictions we impose upon the subgroup of PSL(2,C). In Section 3 we introduce the geometric conditions which must apply for our algorithm to hold, and we prove that these are generically satisfied. We then show in Section 4 that these conditions are in fact sufficient to verify the hypotheses of Poincare's Theorem. In Section 5 we look at the field containing our matrix entries.


OTHER PROJECTS


Last updated: November 2014