The symmetric genus of sporadic groups

Conder, M. D. E.; Wilson, R. A.; Woldar, A. J.

doi:10.1090/S0002-9939-1992-1126192-2

The symmetric genus of sporadic groups
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by M. D. E. Conder, R. A. Wilson and A. J. Woldar

Proc. Amer. Math. Soc. 116 (1992), 653-663

DOI: https://doi.org/10.1090/S0002-9939-1992-1126192-2

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Abstract:

Given a finite group $G$, the symmetric genus of $G$ is defined to be the smallest integer $g$ such that $G$ acts faithfully on a closed orientable surface of genus $g$. Previous to this work, the task of determining the symmetric genus for the sporadic simple groups had been completed for all but nine groups: ${{\text {J}}_3}$, $\operatorname {McL}$, $\operatorname {Suz}$, ${\text {O’N}}$, $\operatorname {Co}_2$, $\operatorname {Fi}_{23}$, $\operatorname {Co}_1$, ${\text {B}}$, and ${\text {M}}$. In the present paper the authors resolve the problem for six of these groups, viz. ${{\text {J}}_3}$, $\operatorname {McL}$, $\operatorname {Suz}$, ${\text {O’N}}$, $\operatorname {Co}_2$, and $\operatorname {Co}_1$. Significant progress is also reported for the group $\operatorname {Fi}_{23}$.

References

Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
John J. Cannon, An introduction to the group theory language, Cayley, Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 145–183. MR 760656
Marston D. E. Conder, Some results on quotients of triangle groups, Bull. Austral. Math. Soc. 30 (1984), no. 1, 73–90. MR 753563, DOI 10.1017/S0004972700001738
Marston D. E. Conder, The symmetric genus of alternating and symmetric groups, J. Combin. Theory Ser. B 39 (1985), no. 2, 179–186. MR 811121, DOI 10.1016/0095-8956(85)90047-4
Marston Conder, Hurwitz groups: a brief survey, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 359–370. MR 1041434, DOI 10.1090/S0273-0979-1990-15933-6
Marston Conder, The symmetric genus of the Mathieu groups, Bull. London Math. Soc. 23 (1991), no. 5, 445–453. MR 1141014, DOI 10.1112/blms/23.5.445
Marston Conder, Random walks in large finite groups, Australas. J. Combin. 4 (1991), 49–57. Combinatorial mathematics and combinatorial computing (Palmerston North, 1990). MR 1129268

The symmetric genus of sporadic groups: announced results

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 4th ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 14, Springer-Verlag, Berlin-New York, 1980. MR 562913
L. Finkelstein and A. Rudvalis, Maximal subgroups of the Hall-Janko-Wales group, J. Algebra 24 (1973), 486–493. MR 323889, DOI 10.1016/0021-8693(73)90122-1
A. Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892), no. 3, 403–442 (German). MR 1510753, DOI 10.1007/BF01443420
Gareth A. Jones and David Singerman, Complex functions, Cambridge University Press, Cambridge, 1987. An algebraic and geometric viewpoint. MR 890746, DOI 10.1017/CBO9781139171915
Stephen A. Linton, The maximal subgroups of the Thompson group, J. London Math. Soc. (2) 39 (1989), no. 1, 79–88. MR 989921, DOI 10.1112/jlms/s2-39.1.79
Stephen A. Linton and Robert A. Wilson, The maximal subgroups of the Fischer groups $\textrm {Fi}_{24}$ and $\textrm {Fi}’_{24}$, Proc. London Math. Soc. (3) 63 (1991), no. 1, 113–164. MR 1105720, DOI 10.1112/plms/s3-63.1.113
A. M. Macbeath, Generators of the linear fractional groups, Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) Amer. Math. Soc., Providence, R.I., 1969, pp. 14–32. MR 0262379
Wilhelm Magnus, Noneuclidean tesselations and their groups, Pure and Applied Mathematics, Vol. 61, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0352287
R. A. Parker, The computer calculation of modular characters (the meat-axe), Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 267–274. MR 760660
Rimhak Ree, A theorem on permutations, J. Combinatorial Theory Ser. A 10 (1971), 174–175. MR 269519, DOI 10.1016/0097-3165(71)90020-3
Chih-han Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42. MR 251216, DOI 10.1007/BF02392383
Leonard L. Scott, Matrices and cohomology, Ann. of Math. (2) 105 (1977), no. 3, 473–492. MR 447434, DOI 10.2307/1970920
David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38. MR 322165, DOI 10.1112/jlms/s2-6.1.29
Thomas W. Tucker, Finite groups acting on surfaces and the genus of a group, J. Combin. Theory Ser. B 34 (1983), no. 1, 82–98. MR 701174, DOI 10.1016/0095-8956(83)90009-6
Andrew J. Woldar, On Hurwitz generation and genus actions of sporadic groups, Illinois J. Math. 33 (1989), no. 3, 416–437. MR 996351
Andrew J. Woldar, Representing $M_{11},\;M_{12},\;M_{22}$ and $M_{23}$ on surfaces of least genus, Comm. Algebra 18 (1990), no. 1, 15–86. MR 1037897, DOI 10.1080/00927879008823902
Andrew J. Woldar, Sporadic simple groups which are Hurwitz, J. Algebra 144 (1991), no. 2, 443–450. MR 1140615, DOI 10.1016/0021-8693(91)90115-O
Andrew J. Woldar, The symmetric genus of the Higman-Sims group HS and bounds for Conway’s groups $\textrm {Co}_1,\textrm {Co}_2$, Illinois J. Math. 36 (1992), no. 1, 47–52. MR 1133769

On the symmetric genus of simple groups

M. F. Worboys, Generators for the sporadic group $\textrm {Co}_{3}$ as a $(2,\,3,\,7)$ group, Proc. Edinburgh Math. Soc. (2) 25 (1982), no. 1, 65–68. MR 648901, DOI 10.1017/S0013091500004144