# Research

The page contains original results obtained by myself with, or without collaborators. Please note that use of this material is only permitted provided that it properly cites this page or the related paper (if published). It is forbidden to disseminate any part of the published material without reference to this page. The conditions for using the published material may be altered by publishers of related papers. As regards the source code published on this page, it may be used in parts or in its entirety in research projects of other people with reference to the authors. Using the source code may also be regulated by licence agreements applying to its parts (modules, libraries, etc.).

## Census of Quadrangle Groups Inclusions

Go to the atlas page. Actual version of census is 2.0. Last modification 1 August 2019.

## Edge transitive maps on orientable surfaces

The webpage contains an original computational output based on the joint work by R. Nedela, J. Karabáš and M. Skyvová: the list of isomorphism classes of edge-transitive maps on the orientable surfaces of genus $$g>1$$. The paragraph became too long for this page, so you can continue reading this story here

## Actions of finite groups on Riemann surfaces of higher genera

The paragraph about the census of actions of discrete groups became too long and I decided to move it into the separate sub-page dedicated to this topic only. To read about the census, click here...

## Archimedean maps

Here is the complete census of Archimedean maps of genera from two to four. Presented data are based on the newest revision my 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"*.
*the work was done in collaboration with Dept. of Mathematics, University of Aveiro, Portugal The census of Archimedean maps follows

 Genus 2 Representatives of iso-classes Full text output catalogue BCK classification Genus 3 Representatives of iso-classes Full text output catalogue BCK classification Genus 4 Representatives of iso-classes Full text output catalogue BCK classification

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.

### References

1. Karabáš, J., and Nedela, R., Archimedean maps of higher genera, Math. Comp. 81 (2012), 569-583.
2. Breda d'Azevedo, A., Catalano, A. D., and Karabáš, J., Two infinite families of Archimedean maps of higher genera (accepted in Ars Combinatoria).

## Actions of cyclic groups on orientable surfaces

Thanks to e-mail communication with Timothy Walsh, Roman Nedela and Alexander Mednykh I decided to publish 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<=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: LCM(m1,m2,...,mr) of branch indexes m1,m2,...,mr equals LCM(m1,m2,...,mi-1,mi+1,...,mr) for every i in 1..r. The numbers of actions of a cyclic group Zn are determined in terms of numbers EpiO1(O), Zn) of order-preserving epimorphisms from a Fuchsian group π1(O) onto a cyclic group Zn, where π1(O) is an orbifold fundamental group of an orbifold O. Since π1(O) is a Fuchsian group, it can be expressed as a group with signature (γ; {m1,m2,…,mr}). 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

 Genera 1..10 Genera 11..20 Genera 21..30 Genera 31..40 Genera 41..50 Genera 51..60 Genera 61..70 Genera 71..80 Genera 81..90 Genera 91..101

I have no plan to continue with generating higher genera. The correctness of the census was independently checked by T. Walsh up to genus 9 (probably he may continue to g=27). If you will discover any problem in lists, please contact me immediately. Please cite this page when using these results in your scientific projects. Thank you.

### References

1. Bujalance, E., Etayo, J. J., Gamboa, J. M., and Gromadzki, G., Automorphism groups of compact bordered Klein surfaces, Springer-Verlag, Berlin, 1990.
2. Cannon, J. J., and Bosma, W. (Eds.) Handbook of Magma Functions, Edition 2.15 (2009).
3. Mednykh, A., and Nedela, R., Enumeration of unrooted maps of a given genus, J. Combin. Theory Ser. B 96 (2006), 706 - 729.

## 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 V. Easson, concerning Thurston's Symmetrization Theorem.