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The Johnson filtration, pseudo-Anosov dilatation, and curve complex translation
Geometry/Topology| Speaker: | Dan Margalit, University of Utah |
| Location: | 2112 MSB |
| Start time: | Wed, Apr 5 2006, 4:10PM |
Description
Let S be a closed orientable surface and let G be its fundamental group. We
consider the smallest dilatation of any pseudo-Anosov homeomorphism of S
acting trivially on G/G_k, the quotient of G by the k-th term of its lower
central series, k > 0. We prove that this minimal dilatation is bounded
above and below, independently of genus, with bounds tending to infinity
with k. For example, in the case of the Torelli group I(S), we prove that
L(I(S)), the logarithm of the minimal dilatation in I(S), satisfies .196 <
L(I(S)) < 4.127. In contrast, we find pseudo-Anosov mapping classes acting
trivially on G/G_k whose asymptotic translation lengths on the complex of
curves tend to zero as genus tends to infinity.