FOUR QUESTIONS ABOUT MAPPING CLASS GROUPS
9
In other words, the subgroups in (1) are "small" in the sense that the set
G\ G2 Gp = {gxg2 gP\gi e Gt}
fails to contain high powers of (f)f for any p and any subgroups G{ as in (1).
Now I would like to propose that certain other subgroups of MCG(Sg) are
small as well. Let q : Sg —* S' be a covering map of degree 1. Define
G(q) = {(j) G MCG(Sg)\(j) is a lift of a mapping class on S'}
PROBLEM 4.1. Show that there are many quasihomomorphisms f:MCG(Sg)-^
R that satisfy (1) and (2) above plus
(3) / is bounded on every G{q).
I remark that the consequence about product sets of G(g)'s was shown to be
true by a different method (Bestvina-Feighn, unpublished)
References
[Akr03] H. Akrout, Singularites topologiques des systoles generalisees, Topology 42 (2003), no. 2,
291-308. MR1 941 437
[BB97] Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups,
Invent. Math. 129 (1997), no. 3, 445-470. MR98i:20039
[BF02a] Mladen Bestvina and Mark Feighn, Proper actions of lattices on contractible manifolds,
Invent. Math. 150 (2002), no. 2, 237-256. MR2004d:57042
[BF02b] , Proper actions of lattices on contractible manifolds, Invent. Math. 150 (2002),
no. 2, 237-256. MR1933585 (2004d:57042)
[BF02c] Mladen Bestvina and Koji Fujiwara, Bounded cohomology of subgroups of mapping class
groups, Geom. Topol. 6 (2002), 69-89 (electronic). MR2003f:57003
[Bie76] Robert Bieri, Normal subgroups in duality groups and in groups of cohomological di-
mension 2, J. Pure Appl. Algebra 7 (1976), no. 1, 35-51. MR52 #10904
[BKK02] Mladen Bestvina, Michael Kapovich, and Bruce Kleiner, Van Kampen's embedding ob-
struction for discrete groups, Invent. Math. 150 (2002), no. 2, 219-235. MR2004c:57060
[Har85] John L. Harer, Stability of the homology of the mapping class groups of orientable
surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215-249. MR87f:57009
[Iva89] N. V. Ivanov, Stabilization of the homology of Teichmuller modular groups, Algebra i
Analiz 1 (1989), no. 3, 110-126. MR91g:57010
[Joh83] Dennis Johnson, The structure of the Torelli group. I. A finite set of generators for T,
Ann. of Math. (2) 118 (1983), no. 3, 423-442. MR85a:57005
[Kir97] Robion Kirby, Problems in low-dimensional topology, Geometric topology (Athens, GA,
1993) (Robion Kirby, ed.), AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc, Prov-
idence, RI, 1997, pp. 35-473. MR1 470 751
[Mes92] Geoffrey Mess, The Torelli groups for genus 2 and 3 surfaces, Topology 31 (1992),
no. 4, 775-790. MR93k:57003
[Mil86] Edward Y. Miller, The homology of the mapping class group, J. Differential Geom. 24
(1986), no. 1, 1-14. MR88b:32051
[MM86] Darryl McCullough and Andy Miller, The genus 2 Torelli group is not finitely generated,
Topology Appl. 22 (1986), no. 1, 43-49. MR87h:57015
[Mor84] Shigeyuki Morita, Characteristic classes of surface bundles, Bull. Amer. Math. Soc.
(N.S.) 11 (1984), no. 2, 386-388. MR85j:55032
[Mor87] , Characteristic classes of surface bundles, Invent. Math. 90 (1987), no. 3, 551-
577. MR89e:57022
[Mum83] David Mumford, Towards an enumerative geometry of the moduli space of curves, Arith-
metic and geometry, Vol. II, Progr. Math., vol. 36, Birkhauser Boston, Boston, MA,
1983, pp. 271-328. MR85j:14046
[MW] lb Madsen and Michael S. Weiss, The stable moduli space of Riemann surfaces: Mum-
ford's conjecture, arXiv, math.AT/0212321.
[SS99] Paul Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces,
J. Differential Geom. 52 (1999), no. 3, 407-452. MR2001d:32018
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