Jeffreys, L. (2021) Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs. Transactions of the American Mathematical Society, 374(8), pp. 5739-5781. (doi: 10.1090/tran/8374)
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Abstract
In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmüller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.
Item Type: | Articles |
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Status: | Published |
Refereed: | Yes |
Glasgow Author(s) Enlighten ID: | Jeffreys, Mr Luke |
Authors: | Jeffreys, L. |
College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
Journal Name: | Transactions of the American Mathematical Society |
Publisher: | American Mathematical Society |
ISSN: | 0002-9947 |
ISSN (Online): | 1088-6850 |
Published Online: | 20 January 2021 |
Copyright Holders: | Copyright © 2021 American Mathematical Society |
First Published: | First published in Transactions of the American Mathematical Society 374(8): 5739-5781 |
Publisher Policy: | Reproduced in accordance with the publisher copyright policy |
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