Seminar of Graph Theory - Martin Mačaj (12.10.2017)

Thursday 12.10.2017 at 9:50, Lecture room M/213

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Martin Mačaj:
On minimal kaleidoscopic regular maps with trinity symmetry

Abstract:
It is known that the monodromy group of any regular unoriented map ${\cal M}$ can be uniquely (up to isomorphism) represented as an abstract group $G$ with a triple $(a,b,c)$ of involutory generators such that $ac=ca$. In this representation operators of duality, Petrie-duality and $e$th-hole operator can be interpreted as the change of the generating triple to the triple $(c,b,a)$, $(ac,b,c)$ and $(a,(bc)^{e-1},c)$, respectively. Regular map ${\cal M}$ which is invariant with respect to all of the above operators (for $e$ co-prime to the valency of ${\cal M}$) is said to be {\em kaleidoscopic regular map with trinity symmetry (KRT)}. We say that a  KRT is {\em minimal} if it covers no KRT besides itself and the trivial map. In this talk we recall how the algebraic description of regular maps and their operators can be obtained. Further, we characterize minimal KRTs as least common covers of special families of regular maps with simple monodromy groups. We also presents results of exhaustive computer search with focus on KRTs of odd valency and KRTs with simple monodromy groups.

 

 

 

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