Contact me:

Institue of Mathematics
Matej Bel University
Ďumbierska 1
Banská Bystrica
SK-97411
Slovak republic

My e-mail

Contents:

• Publications
• Conferences
• 3-manifolds
• Archimedean solids
• Finite groups actions
• Cyclic orbifolds

Last modified:

Research

Actions of finite groups on Riemann surfaces of higher genera

This is a complete census of actions of finite groups on Riemann surfaces of higher genera. At this moment (November 2, 2011) the census is complete up to genus 24. The previous (similar) censa were published in the papers of Broughton (genus 2 and 3), Bogopolski (genus 4) and Kuribayashi and Kimura (genus 5). The census of all actions for genera 2 to 15 was bublished here for several years. The list below contains also these genera as well - just for check. Moreover, I provide the number of the actions of a finite group G on the respective surface S_g of genus g.

The process itself is the application of classical results from low-dimensional topology and group theory, i.e., the finite group G acting on S_g is nothing but the quotient of the Fuchsian group F(γ; {m₁,...,m_r}) ≅ π₁(S_g/G) such that the epimorphism F->G has torsion-free kernel. The quotient space S_g/G is an orbifold with fixed signature. The order of G is bounded for g>1 by Hurwitz bound 84(g-1). The computation is done in two steps. At first all possible numerical solutions of Riemann-Hurwitz equation for coverings of surface S_g were computed. The possible signatures of orbifolds with r branch-points (ramification points) (γ; {m₁,...,m_r}) were obtained. Note that for every genus g>1 the number r is bounded by 2g+2. The second step, obtaining existing actions, is more complicated. Groups were examined according to their respective order. Quotients of the respective Fuchsian groups were computed (constructed) using LowIndexNormalSubgroups (resp. using HomomorphismsProcess) procedure contained in Magma. Moreover, the number of respective homomorphisms F->G were added, where it was possible to enumerate them.

The census is divided into files, one file per genus. The files are ordinary text files with Unix line ends, one recore per line. The format of the files is the following:

In the header are there is written the constraint of search for the number of actions of the given signature (usually 100000, but may be 50000 in larger genera). The record consist of

1. The unique identifier of the record in the form "Og.n", where g is the genus of the surface upstairs and n is the position in the file;
2. The respective signature, in the shortened form, i.e. ...,2,2,2,.... is written as ...,2^3,....;
3. The code of the group G, as recorded in the order of Small Groups Library, in the form <o,p>, where oStands for the order of G and p is the position in the respective catalogue. This code is valid for Magma and GAP, respectively;
4. The number of epimorphisms found. If quotation mark appears, then the number of epimorphisms exceeded the limit (as in header). Note that in case of cyclic group you may found the number of epimorphisms using different methods (see here, for example);
5. Following the semicolon, there is a structural description of G. This description can be obtained by calling StructuralDescription in GAP.

Here are the catalogues:

(^*) the computation was not finished due to problems with 2-groups. Refer M. Conder's list of regular reflexible and chiral maps or list of triangluar and quadrangular actions for the existence of actions. Will be fixed in the next version.

References

1. Bogopolski, O. V., Classifying the actions of finite groups on orientable surfaces of genus 4, Siberian Adv. Math. 7 (1997), pp. 9-38
2. Broughton, S. A., Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991), pp. 233-270.
3. Bujalance, E., Etayo, J. J., Gamboa, J. M. and Gromadzki, G., Automorphism groups of compact bordered Klein surfaces, Springer-Verlag, Berlin, 1990.
4. Cannon, J. J. and Bosma, W. (Eds.) Handbook of Magma Functions, Edition 2.13 (2006), 4350 pages.
5. Conder, M., List of orientable regular maps on surfaces of genus 2 to 101, (http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt).
6. Conder, M., List of chiral orientably-regular maps on surfaces of genus 2 to 101, (http://www.math.auckland.ac.nz/~conder/ChiralMaps101.txt).
7. Conder, M., List of all large groups of automorphisms of compact Riemann surfaces of genus 2 to 101, up to equivalence of the group action, listed by genus, (http://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt).
8. The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4 ; 2006, (http://www.gap-system.org).
9. Karabáš, J. and Nedela, R., Archimedean solids of higher genera (submitted)
10. Kuribayashi, A. and Kimura, H., Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), pp. 80-103.
11. Mednykh, A. and Nedela, R., Enumeration of unrooted maps of a given genus, J. Combin. Theory Ser. B 96 (2006), 706 - 729.

Archimedean solids

^* the work was done in collaboration with Dept. of Mathematics, University of Aveiro, Portugal The census of Archimedean solids follows

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. Archimedean maps of higher genera (with R. Nedela), Math. Comp. 81 (2012), 569-583.
2. Two infinite families of Archimedean maps of higher genera (with A. Breda and D. Catalano)(submitted).

Actions of cyclic groups over 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 over 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,...,m_r) of branch indexes m₁,m₂,...,m_r equals LCM(m₁,m₂,...,m_i-1,m_i+1,...,m_r) for every i in 1..r. 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 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.

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

Census

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.

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

List of Publications

1. 6-decomposition of snarks (with E. Máčajová and R. Nedela)(accepted in European Journal of Combinatorics)
2. Two infinite families of Archimedean maps of higher genera (with A. Breda and D. Catalano)(submitted).
3. Archimedean maps of higher genera (with R. Nedela), Math. Comp. 81 (2012), 569-583.
4. Archimedean solids of genus two (with R. Nedela), Electronic Notes in Discrete Mathematics 28 (2007), 331-339.
5. Three-manifolds with Heegaard genus at most two represented by crystallisations with at most 42 vertices (with P. Maličký and R. Nedela), Discrete Mathematics 307 (2007), 2569 - 2590.
6. 3-manifolds with Heegaard genus at most two, PhD Thesis, Mathematical Institute of SAS, Bratislava, 2005.
7. Fundamental groups of prime 3-manifolds of genus at most two (with P. Maličký and R. Nedela), Tatra Mountains Math. Publ. 36 (2007), 1-11, Proceedings of conference "Graphs 2004".
8. Minimal Representatives of G-classes of 3-Manifolds of Genus Two (with R. Nedela), Acta Univ. M. Belii. Ser. Mathematics 10 (2003), 21-42.

Conferences, seminars, lectures