Abstract
In this paper we generalise Singerman’s results on triangle group inclusions to the broader class of generalised quadrangle groups, that is, Fuchsian groups with signature of genus 0 and generated by three or four elliptic generators. For any possible inclusion \(P<Q\) we also give the number of non-conjugate subgroups of Q isomorphic to P.
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Acknowledgments
This work was supported by the Research and Development Cooperation project Slovakia–Portugal, ID number: SK-PT-NEWPROJECT-12522, supported by the portuguese side by the Portuguese Foundation for Science and Technology FCT (Fundação para a Ciência e a Tecnologia). The work of the first two authors was partially supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications (University of Aveiro) and FCT within project PEst-OE/MAT/UI4106/2014. The work of of the third and the fourth author was supported by the Ministry of Education of Slovak Republic, Grant VEGA 1/0150/14, Grant APVV-SK-PT-0004-12, and by the Project L01506 of the Czech Ministry of Education, Youth and Sports.
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Appendix: List of inclusions of generalised quadrangle groups
Appendix: List of inclusions of generalised quadrangle groups
The following tables display the census of inclusions of generalised quadrangle groups. In both tables we have decided to include non-realisable numerical solutions as well. These have no label and the number of realisations is not assigned (crossed off column). The labels of realisable inclusions correspond to those used in d’Azevedo et al. (2016a, (2016b). The meaning of labels is: “lower case letter”—an infinite family of normal inclusions as in Singerman (1972), “upper case letter”—a sporadic inclusion as in Singerman (1972), “f” and number—an infinite family of normal inclusions of generalised quadrangle groups, “F” and number—an infinite family of inclusions of generalised quadrangle groups, and “S” and number—a sporadic inclusion of generalised quadrangle groups. According to the discussion at the end of Sect. 5 we did not include the inclusions which are embedded in some way in an infinite family. The inclusions are ordered first by index and then by lexicographic order of the parameters of the upper group. Groups are taken as abstract groups, hence the parameters of presentations were ordered lexicographically such that constants were lexicographically before polynomials. The corresponding passport is ordered in the same order as the parameters of the upper group are ordered. The number of realisations of an inclusions (or of a member of an infinite family) is written in the way that the first number is the total number of realisations, the second is the number of reflexible realisations, and the last number is the number of chiral realisations (hence, always even).
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d’Azevedo, A.B., Catalano, D.A., Karabáš, J. et al. Quadrangle groups inclusions. Beitr Algebra Geom 58, 369–394 (2017). https://doi.org/10.1007/s13366-016-0309-3
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DOI: https://doi.org/10.1007/s13366-016-0309-3