Warren Dicks and Edward Formanek,
Automorphism subgroups of finite index in algebraic mapping class groups.
J. Algebra 189 (1997), 58-89.
Addenda
October 2, 2003
Vladimir Shpilrain has drawn our attention to the following article which
we overlooked.
H. Zieschang, "A note on the mapping class groups of surfaces and planar
discontinuous groups,"
pages 206-213 in "Low-dimensional topology (Chelwood Gate, 1982)", ed.
by R. Fenn,
London Math. Soc. Lecture Note Ser., Vol. 95,
Cambridge Univ. Press, Cambridge, 1985.
This article extends, to (possibly unorientable) surfaces with finitely
many punctures and multiple points,
the topological proof due to J. Birman, L. Bers, and
C. Maclachlan of the results for orientable surfaces
with finitely many punctures, which we state as Theorem 4.9 and Remark
4.10.
At the end of Section 1, the author says the results can be "proved using
only combinatorial
group-theoretical arguments", by a "long" proof, that "consists of
many case considerations".
It seems therefore that we can no longer claim priority in this matter,
for the case that we consider.
However, it may be that our proof is simpler.
Errata
November 14, 2007
On page 62, lines 10 and 18, results of Dehn and Epstein, respectively,
are misquoted. If
(g,n) = (0,0) or (0,1), then the fundamental group of the
(g,n)-surface is trivial, but orientation-
reversing maps of the (g,n)-surface are not isotopic to
the identity; here the topological mapping
class group has order two, while the algebraic mapping class group is
trivial and should therefore
be redefined to have order two, in these two cases. With this
new definition, Theorem 4.9 would
have no exceptions.
On page 63, delete line 16, that is, the third paragraph.
On page 66, at the end of the 8th last line, add a closing ].
On page 76, lines 12-13, the statement of Proposition 4.7 is not
correct, since a free group is
its own universal central extension. There are three cases, corresponding
to Sigmag,n being
abelian, or a nonabelian surface group, or a nonabelian free group:
(a) If 2g + n -1 \le 1, then Sigmag,n
/Ctr and \hat Sigmag,n+1
are both trivial.
(b) If n = 0 and 2g + n -1 \ge
2 (that is, g \ge 2) , then \hat Sigmag,1
is the universal central
extension of the (nonabelian) surface group
Sigmag,0 = Sigmag,0
/Ctr .
(c) If n \ge 1 and 2g + n -1 \ge 2, then
Sigmag,n = Sigmag,n /Ctr
is a (nonabelian) free group
of rank 2g + n -1, and
\hat Sigmag,n+1 is isomorphic
to the direct product of Sigmag,n and
an infinite cyclic group.
On page 77, line 6, change "Sigmag,N+1"
to "Sigmag,n+1 ".
On page 77, line 12, change "Corollary 5.4.3" to "Theorem 5.7.2".
On page 77, lines 16-17, Change "the inner automorphisms lies" to "the
inner automorphisms and
the braid automorphisms lie". (Recall that the braid automorphisms
are sigmai, ( i = 1, ..., n),
where sigmai sends zito zi+1,
sends zi+1 to zi+1-1zizi+1,
and fixes all the other generators.)
On page 88, line 11, change "other" to "outer".
On page 88, on the 6th last line, add "pp. 199-270 in "Topology
and geometry-Rohlin Seminar",
ed. by O. Ya. Viro," before "Lecture Notes".
On page 89, line 2, change "(1985)" to "(1975)".
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