Characters and Automorphism Groups
of Compact Riemann Surfaces
Lehrstuhl D für Mathematik RWTH, 52056 Aachen, Germany
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
(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:
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.
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
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 +
( rhoG - 1<xi>G ) .
For any character chi of G, the scalar product with this character
is clearly nonnegative, thus
2 ·[ chi, rhoG - 1G ] <=
[ chi, rhoG - 1<xi>G ] .
Because of [ chi, rhoG ] = chi(1) and together with
Frobenius reciprocity, this implies
2 ·( chi(1) - [ chi, 1G ] ) <=
( 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.)