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Coxeter-Petrie complexes and groups
by
David B. Surowski
Kansas State University
Coauthors: Kevin Anderson
Coxeter-Petrie Complexes naturally arise as thin diagram geometries whose rank three residues contain all of the dual forms of an algebraic map (= combinatorial Riemann surface). Corresponding to an algebraic map is its classical dual, which is obtained simply by interchanging the vertices and faces, as well as its Petrie dual, which comes about by replacing the faces by the so-called Petrie polygons. Jones and Thornton have shown that these involutory duality operations generate the symmetric group S3, giving in all six dual forms, and whose source is as the outer automorphism group of the infinite triangle group generated by involutions s1, s2, s3, subject to the additional relation s1s3 = s3s1. This outer automorphism group acts transitively on the involutions s1, s3, s1s3 and fixes s2. If we allow these four involutions to define the nodes of a Coxeter diagram of type D4 (but with edges marked with \infty), then there is a natural extension from the original algebraic map to a thin Coxeter complex of rank 4. These are fully explicated in case the original algebraic map is a Platonic map.
Date received: February 22, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagg-08.