Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

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
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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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
172865EPSRC DTP 16/17 and 17/18Tania GalabovaEngineering and Physical Sciences Research Council (EPSRC)EP/N509668/1Research and Innovation Services