19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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
Co-authors
- Prof. Dan ARCHDEACON University of Vermont
- Prof. Jozef SIRAN Open University, UK
- Dr. Young Soo KWON Yeungnam University, Korea