for

Characters and Automorphism Groups
of Compact Riemann Surfaces

by

Thomas Breuer

Lehrstuhl D für Mathematik
RWTH, 52056 Aachen, Germany
E-mail: sam@math.rwth-aachen.de

 Errata
p. 6, l. 17:
Write "The map" not "So the map".
p. 8, l. -14:
Insert a comma before cr.
p. 20, l. -4 to -1:
Theorems 3A and 3B in [Gre63] are not correct, so replace the theorem by the following:
THEOREM 5.1 ([Sin72], Theorems 1 and 2]). No Fuchsian group with signature (g; m1, m2, ..., mr) is finitely maximal if and only if the signature is one of
 (0;2,n,2n), (0;3,n,3n), (0;m,m,n), (0;m,m,n,n), (1;n,n), (1;n), (2;-) .
(The error does not affect the material in the rest of the book.)
p. 26, l. 6:
Add a period at the end of the sentence.
p. 52, l. 6:
Insert "at least" after "has".
p. 66, l. -9 to -6:
The formulation of Lemma 17.6 may be misleading, so replace it by the following:
LEMMA 17.6. Let Gamma be a Fuchsian group, and m a positive integer such that no group of order m is perfect. If Gamma has a surface kernel factor of order m then there is a prime p dividing gcd(m, [Gamma : Gamma']) and a normal subgroup of index p in Gamma whose signature is admissible for m/p.
p. 80, l. -11:
Add a closing bracket after 9E.
p. 100, l. -1 and p. 112, l. 3:

 Write Tr( Phi ) + Tr( Phi ) instead of Phi + Phi

.
p. 118, l. -11 to -9:
Remove this paragraph.
p. 176, l. 8 to 9:
Remove "if we require 1 <= Np,1 <= Np,2 <= ... <= Np,sp".
p. 169, l. 7:
Add that p >= 7 is required, since for p = 3, chi is not a proper character. (Note that Lemma 34.8 holds also for p = 3.)
p. 189, l. -1:
Write det( Dm ) = 0.
p. 194, l. -14:
Add the reference [Sin72], which (after the above correction) is referred to on p. 20.

p. 62 f.:
The (2C, 3D, 8C)-generation of the group Fi23 established in Section 16 with character-theoretic methods has been proved by Robert A. Wilson, via explicit computations with the group Fi23.
He has computed also the (strong) symmetric genera of the Baby Monster and the Monster. For the Baby Monster, it arises from (2,3,8)-generation. The Monster is a Hurwitz group.
For details, see [Wil93,Wil97,Wil01].
p. 98, l. 5 to 7:
For a character that comes from a Riemann surface, the representation of the sum with its complex conjugate in terms of permutation characters has been derived also by A. J. Broughton; in [Bro90], this is used to prove Corollary 15.10 in an alternative way, which can be rephrased in our terminology, as follows.
Suppose that the elements x1, x2, ..., xr with the property x1 x2 ... xr = 1 generate the group G. This gives rise to a surface kernel epimorphism Phi: Gamma(0;|x1|, |x2|, ... |xr|) -> G, with induced character Tr( Phi ). By Corollary 22.5, we have
 Tr( Phi ) + Tr( Phi ) = 2 ·1G - 2 · rhoG + rSum i=1 ( rhoG - 1G ) .
For any character chi of G, the scalar product with this character is clearly nonnegative, thus
 2 ·[ chi, rhoG - 1G ] <= rSum i=1 [ chi, rhoG - 1G ] .
Because of [ chi, rhoG ] = chi(1) and together with Frobenius reciprocity, this implies
 2 ·( chi(1) - [ chi, 1G ] ) <= rSum i=1 ( chi(1) - [ chi, 1 ] ) .
(In [Bro90], this is in fact stated also for the case that the preimage of Phi has positive orbit genus. But then the analogon of the above condition is trivially satisfied.)

## References

[Bro90]
A. J. Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1990), 233-270.
[Gre63]
L. Greenberg, Maximal Fuchsian Groups, Bull. Amer. Math. Soc. 69 (1963), 569-573, MR 26, 6127.
[Sin72]
D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29-38.
[Wil93]
R. A. Wilson, The Symmetric Genus of the Baby Monster, Quart. J. Math. Oxford Ser. (2) 44 (1993), 513-516.
[Wil97]
R. A. Wilson, The Symmetric Genus of the Fischer group Fi23, Topology 36 (1997), 379-380.
[Wil01]
R. A. Wilson, The Monster is a Hurwitz group, J. Group Theory 4 (2001), 367-374.

Last update July 24th, 2008.

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