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Growth, projections and bounded generation of mapping class groups

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Jean Sun (Yale)

Abstract:
We investigate the non-bounded generation of subgroups of mapping class groups
through the hierarchy in curve complexes developed by Masur and Minsky
(2000). We compare the subsurface projections to nearest point projections in
curve complexes and extend Behrstock's inequality to include geodesics in
curve complexes of subsurfaces in the Inequality on Triples in
Bestivina-Bromberg-Fujiwara (2010). Based on this inequality, we can estimate
translation lengths of words in the form g_1^{n_1}... g_k^{n_k}
when Σ |n_k| is sufficiently large for any given sequence
(g_i)_1^k in a mapping class group. With a growth argument, we further show
that any subgroup of a mapping class group is boundedly generated if and only
if it is virtually abelian.