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true by a different method (Bestvina-Feighn, unpublished)
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