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Licensed Unlicensed Requires Authentication Published by De Gruyter July 8, 2014

The regular Grünbaum polyhedron of genus 5

  • G. Gévay EMAIL logo , E. Schulte and J. M. Wills
From the journal Advances in Geometry

Abstract

We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary space E3. This polyhedron was originally discovered by Grünbaum in 1999, but was recently rediscovered by Brehm andWills. We establish isomorphism of the Grünbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grünbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g ≥ 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g ≥ 2.

Published Online: 2014-7-8
Published in Print: 2014-7-1

© 2014 by Walter de Gruyter Berlin/Boston

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