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true by a different method (Bestvina-Feighn, unpublished)

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