9-13 June 2013
Koper, Slovenia
UTC timezone
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Almost totally branched covers between regular maps

Presented by Prof. Roman NEDELA
Type: Oral presentation
Track: Symmetries in Graphs, Maps and Other Discrete Structures

Content

A map is a $2$-cell decomposition of a closed surface. A map on an orientable surface is called orientably regular if its group of orientation-preserving automorphisms acts transitively on the set of darts (edges endowed with an orientation). We investigate orientably regular maps with an unfaithful action of the automorphism group simultaneously at vertices and faces. In particular, we say that a covering $p\colon\cal{N}\to \cal{M}$ between orientably regular maps is \emph{ almost totally branched} if its group of covering transformations is a product $XY$, $X\leq\Fix_{\cal{N}}(F)$, $Y\leq\Fix_{\cal{N}}(V)$, where $\Fix_{\cal{N}}(F)$ and $\Fix_{\cal{N}}(V)$ fix pointwisely the face-centres and the vertices of $\cal{N}$, respectively. We first show that every branched covering $p\colon {\cal{N}}\to {\cal{M}}$ can be written as a composition $p=p_1\circ p_2\circ p_3$, where $p_3$ is smooth and $p_2$ is almost totally branched. Next we determine the structure of the covering transformation group $G$ of an almost totally branched covering. Three cases can happen, either $G$ is nonabelian containing an abelian subgroup of index two, or it is a product of two cyclic groups, or $G$ is cyclic. In the nonabelian case we dermine the presentation of $G$. As an application we investigate orientably regular maps which are regular covers over platonic maps with a cyclic group of covering transformations. We describe all such maps in terms of parametrised group presentations. This generalises the work of Jones and Surowski [Cyclic regular coverings of the Platonic maps, {\it European J. Combin.} {\bf21} (2000), 333--345] classifying the cyclic regular coverings over platonic maps with branched points exclusively at vertices, or at face-centres.

Place

Location: Koper, Slovenia

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