A179982 Values of the genus g for which there exists a compact Riemann surface of genus g admitting an automorphism group of order 84(g-1), the maximum possible, also known as the Hurwitz bound.

3, 7, 14, 17, 118, 129, 146, 385, 411, 474, 687, 769, 1009, 1025, 1459, 1537, 2091, 2131, 2185, 2663, 3404, 4369, 4375, 5433, 5489, 6553, 7201, 8065, 8193, 8589, 11626, 11665

Offset

1,1

Comments

It is known that the sequence is infinite, and also that the sequence of g for which the Hurwitz bound is not attained is infinite. No generating formula is known.

The group in question is a Hurwitz group, and so is generated by x,y with x^2 = y^3 = (xy)^7 = 1.

For k in A343821 (not in A343822), k!/168 + 1 is in this sequence since the alternating group A_k is a Hurwitz group. In particular, k!/168 + 1 is a term for all k >= 168. - Jianing Song, Jul 13 2021

References

Marston Conder, Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 359-370.

Gareth A. Jones and David Singerman, Complex Functions: An algebraic and geometric viewpoint, Cambridge University Press, 1987, pp. 263-266.

E. B. Vinber and O. V. Shvartsman, Riemann surfaces, Journal of Mathematical Sciences, 14, #1 (1980), 985-1020. Riemann surfaces, Algebra, Topologiya, Geometriya, Vol. 16 (Russian), pp. 191-245, 247 (errata insert), VINITI , Moscow, 1978.

Links

Table of n, a(n) for n=1..32.

Jeffrey M. Cohen, On Hurwitz extensions by PSL_2(7), Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 3, 395-400.

Marston Conder, The genus of compact Riemann surfaces with maximal automorphism group, J. Algebra 108 (1987), no. 1, 204-247.

Chih-han Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969) 13-42.

Wikipedia, Hurwitz Group

Example

g=3 is satisfied by the Klein quartic x^3 * y + y^3 * z + z^3 * x = 0; the group is isomorphic to PSL(2,Z_7), the projective special linear group of 2 X 2 matrices with entries modulo 7.

Prog

(Magma) G<a, b>:=Group<a, b|a^2, b^3, (a*b)^7>; L:=LowIndexNormalSubgroups(G, 86016); for j in L do print (j`Index)/84+1; end for; // Bradley Brock, Nov 25 2012

Crossrefs

Cf. A343821, A343822, A346293.

Sequence in context: A143630 A342732 A218877 * A184816 A310267 A310268

Adjacent sequences: A179979 A179980 A179981 * A179983 A179984 A179985

Keyword

more,nonn

Author

Richard Chapling (rc476(AT)cam.ac.uk), Aug 04 2010

Extensions

Corrected by Bradley Brock, Oct 01 2012

Entry revised by N. J. A. Sloane, Nov 25 2012

Status

approved