# Bled'11 - 7th Slovenian International Conference on Graph Theory

19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
Home > Timetable > Contribution details
PDF | XML

# Regular maps admitting all Wilson operators

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

## Content

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.

## Place

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

More

More