Errata et Addenda

p. 6, l. 17:

Write "The map" not "So the map".

p. 8, l. -14:

Insert a comma before c_r.

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, ..., m_r) 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 <= N_p,1 <= N_p,2 <= ... <= N_p,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( D_m ) = 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, ..., x_r with the property x₁ x₂ ... x_r = 1 generate the group G. This gives rise to a surface kernel epimorphism Phi: Gamma(0;|x₁|, |x₂|, ... |x_r|) -> G, with induced character Tr( Phi ). By Corollary 22.5, we have

Tr( Phi ) + Tr( Phi )   = 2 ·1G - 2 · rhoG + r Sum i=1  ( rhoG - 1<xi>G ) .

For any character chi of G, the scalar product with this character is clearly nonnegative, thus

2 ·[ chi, rhoG - 1G ] <= r Sum i=1  [ chi, rhoG - 1<xi>G ] .

Because of [ chi, rho_G ] = chi(1) and together with Frobenius reciprocity, this implies

2 ·( chi(1) - [ chi, 1G ] ) <= r Sum i=1  ( chi(1) - [ chi<xi>, 1<xi> ] ) .

(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.