Barry Monson - Realizations of regular toroidal maps

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Barry Monson - Realizations of regular toroidal maps

	BARRY MONSON, Department of Mathematics and Statistics, University of New Brunswick-Fredericton, Fredericton, New Brunswick E3B 5A3, Canada
	Realizations of regular toroidal maps

A regular abstract polytope ${\cal P}$ is a poset having the essential structural features of the face lattice of a regular convex polytope, including transitivity of $\textrm{Aut}({\cal P})$ on flags. Other examples include the regular tessellations, star polyhedra and maps on compact surfaces. (Actually, ${\cal P}$ need not be a lattice or have a particularly nice geometric realization.)

Indeed, McMullen (1989) has developed the basic theory of realizations for ${\cal P}$ , basically by using geometric methods to describe the real representations for $\textrm{Aut}({\cal P})$ .

Asia Ivic Weiss and I have recently examined the case that ${\cal P}$ is the toroidal map $\{4,4\}_{(b,0)}$ . In a pretty and unexpected way, the data for the realizations (and group representations) are encoded in a simple picture of the map.

eo@camel.math.ca