On translation lengths of Anosov maps on the curve graph of the torus

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We show that an Anosov map has a geodesic axis on the curve graph of the torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichmuller space, and (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length .
Publisher
SPRINGER
Issue Date
2021-09
Language
English
Article Type
Article
Citation

GEOMETRIAE DEDICATA, v.214, pp.399 - 426

ISSN
0046-5755
DOI
10.1007/s10711-021-00622-1
URI
http://hdl.handle.net/10203/287643
Appears in Collection
MA-Journal Papers(저널논문)
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