Archimedean maps

In collaboration with Dept. of Mathematics, University of Aveiro, Portugal

Here is the complete census of Archimedean maps of genera from two to four. Presented data are based on the newest revision the paper "Archimedean solids of higher genera" which I wrote with Roman Nedela. BCK classification is the listing of classes of Archimedean maps with regard to the paper "Two infinite families of Archimedean maps of higher genera"

Please read the description of the format of the text form of catalogues presented here. The Magma internal format may be useful for people which may want to track the computations which led to the catalogues. The list of representalives of isomorphism classes is added. This list is more convenient for working with than the full output catalogue.

Enumeration of actions of cyclic groups on orientable surfaces

Thanks to Timothy Walsh, Roman Nedela, and Alexander Mednykh

Here is a long list of actions of cyclic groups on orientable surfaces. The result is of crucial importance in map and graph enumeration problems. The procedure for obtaining the result is quite simple. The classification procedure is based on a result of Harvey: "The maximal order \(n\) of a cyclic group acting over an orientable surface is bounded; \(n\leq 4g + 2\), where \(g\) is genus of the surface". Another important fact can be stated as: "The set of branch indexes of the respective orbifold should have the elimination property, if group acting on the surface is abelian". The elimination property reads as follows: \(\operatorname{LCM}(m_1,m_2,\ldots,m_r)\) of branch indexes \(m_1,m_2,\ldots,m_r\) equals \(\operatorname{LCM}(m_1,m_2,\ldots,m_{i-1},m_{i+1},\ldots,m_r)\) for every \(i\in 1\dots r\). The numbers of actions of a cyclic group \(\mathbb{Z}_n\) are determined in terms of an arithmetic function \(\operatorname{Epi}_0(\pi_1(\mathcal{O}), \mathbb{Z}_n)\) of order-preserving epimorphisms from a Fuchsian group \(\mathrm{F}(\gamma;m_1,m_2,\ldots,m_r)\cong \pi_1(\mathcal{O})\) onto a cyclic group \(\mathbb{Z}_n\).

Combining these facts and employing (my) Magma program I have obtained useful tool and the list of actions of cyclic groups on surfaces of genera from 1 to 101 is included here. The program can be used for higher genera as well.

I have prepared the result in semi-colon separated form (raw ASCII text file). This list of coverings may be useful for direct use in computer; it is still human-readable. You can download the file here. The census formatted for reading is below. Be careful, documents may be quite long: the last one has 4006 pages

Census

Classification of 3-manifolds of Heegard genus at most 2

Here you can find full text of my PhD thesis including an explanatory appendix by Vivien Easson, concerning Thurston's Symmetrization Theorem.

• 3-Manifolds with Heegaard Genus at Most Two, PhD. thesis
• Errata to the thesis
• Geometrization of 3-manifolds with Heegaard genus two, appendix to thesis, originally published by V. Easson