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The lower central series and pseudo-Anosov dilatations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Farb, Benson Leininger, Christopher J. Margalit, Dan |
| Copyright Year | 2006 |
| Abstract | The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface Sg of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of Sg tends to zero at the rate 1/g. We consider here the smallest dilatation of any pseudoAnosov homeomorphism of Sg acting trivially on Γ/Γk, the quotient of Γ = π1(Sg) by the k term of its lower central series, k ≥ 1. In contrast to Penner’s asymptotics, we prove that this minimal dilatation is bounded above and below, independently of g, with bounds tending to infinity with k. For example, in the case of the Torelli group I(Sg), we prove that L(I(Sg)), the logarithm of the minimal dilatation in I(Sg), satisfies .197 < L(I(Sg)) < 4.127. In contrast, we find pseudo-Anosov mapping classes acting trivially on Γ/Γk whose asymptotic translation lengths on the complex of curves tend to 0 as g → ∞. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.math.uiuc.edu/~clein/dil.pdf |
| Alternate Webpage(s) | https://faculty.math.illinois.edu/~clein/dil.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0603675v2.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Arabic numeral 0 Class Dilate procedure Fuchsian group Genus Linear algebra Logarithm Pseudo brand of pseudoephedrine Smallest penner |
| Content Type | Text |
| Resource Type | Article |
