Asymptotic translation lengths and normal generation for pseudo-Anosov monodromies of fibered 3-manifolds

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Let M be a hyperbolic fibered 3-manifold. We study properties of sequences (San, ran) of fibers and monodromies for primitive integral classes in the fibered cone of M. The main object is the asymptotic translation length & POUND;C(ran) of the pseudo-Anosov monodromy ran on the curve complex. We first show that there exists a constant C > 0 depending only on the fibered cone such that for any primitive integral class (S, r) in the fibered cone, & POUND;C( r) is bounded from above by C/j & chi;(S)j. We also obtain a moral connection between & POUND;C( r) and the normal generating property of r in the mapping class group on S. We show that for all but finitely many primitive integral classes (S, r) in an arbitrary 2-dimensional slice of the fibered cone, r normally generates the mapping class group on S. In the second half of the paper, we study if it is possible to obtain a continuous extension of normalized asymptotic translation lengths on the curve complex as a function on the fibered face. An analogous question for normalized entropy has been answered affirmatively by Fried and the question for normalized asymptotic translation length on the arc complex in the fully punctured case has been answered negatively by Strenner. We show that such an extension in the case of the curve complex does not exist in general by explicit computation for sequences in the fibered cone of the magic manifold.
Publisher
GEOMETRY & TOPOLOGY PUBLICATIONS
Issue Date
2023-06
Language
English
Article Type
Article
Citation

ALGEBRAIC AND GEOMETRIC TOPOLOGY, v.23, no.3, pp.1363 - 1398

ISSN
1472-2747
DOI
10.2140/agt.2023.23.1363
URI
http://hdl.handle.net/10203/310551
Appears in Collection
MA-Journal Papers(저널논문)
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