19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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Regular maps admitting all Wilson operators

Presented by Prof. Marston CONDER
Type: Oral presentation
Track: Group Actions


A map is a $2$-cell embedding of a connected graph or multigraph in a surface, and is called (fully) {\em regular\/} if its automorphism group has a single orbit on incident vertex-edge-face triples. A map on an orientable surface is said to have {\em trinity symmetry\/} if it is isomorphic to both its dual and its Petrie dual. In that case it admits all six of the so-called {\em standard Wilson operators}. Similarly, given a $d$-valent map $M$, we may define the power map $M^e$ for any $e$ coprime to $d$ by taking $M$ and replacing the cyclic rotation of edges at each vertex on the surface with the $e\,$th power of that rotation. Then a map is said to be {\em kaleidoscopic\/} if it is isomorphic to all of its power maps, and in that case, it admits all of the {\em Wilson `hole' operators}. In this talk, we prove the existence of $d$-valent kaleidoscopic regular maps with trinity symmetry for all even $d$, verifying a conjecture made by Steve Wilson 35 years ago. In some sense, these are the ultimate `supersymmetric maps'. Also we show that the group generated by the Wilson operators can be arbitrarily large.


Location: Bled, Slovenia
Address: Best Western Hotel Kompas Bled

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