The Regular Grunbaum Polyhedron of Genus 5

We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary 3-space. This polyhedron was originally discovered by Grunbaum in 1999, but was recently rediscovered by Brehm and Wills. We establish isomorphism of the Grunbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grunbaum 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.

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