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 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; m_{1}, m_{2}, ..., m_{r}) 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:
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.
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.
Addenda
p. 62 f.:
The (2C, 3D, 8C)-generation of the group Fi_{23}
established in Section 16 with character-theoretic methods
has been proved by Robert A. Wilson,
via explicit computations with the group Fi_{23}.
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 x_{1}, x_{2}, ..., x_{r} with the
property x_{1} x_{2} ... x_{r} = 1 generate the group G.
This gives rise to a surface kernel epimorphism
Phi: Gamma(0;|x_{1}|, |x_{2}|, ... |x_{r}|) -> G,
with induced character Tr( Phi ).
By Corollary 22.5, we have
Tr( Phi ) +
Tr( Phi )
= 2 ·1_{G} - 2 · rho_{G} +
r Sum
i=1
( rho_{G} - 1_{<xi>}^{G} ) .
For any character chi of G, the scalar product with this character
is clearly nonnegative, thus
2 ·[ chi, rho_{G} - 1_{G} ] <=
r Sum
i=1
[ chi, rho_{G} - 1_{<xi>}^{G} ] .
Because of [ chi, rho_{G} ] = chi(1) and together with
Frobenius reciprocity, this implies
2 ·( chi(1) - [ chi, 1_{G} ] ) <=
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.)