@article{742df51847d84dde8aef107e3aa1e155,
title = "Rigidity of the saddle connection complex",
abstract = "For a half-translation surface (Formula presented.), the associated saddle connection complex (Formula presented.) is the simplicial complex where vertices are the saddle connections on (Formula presented.), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism (Formula presented.) between saddle connection complexes is induced by an affine diffeomorphism (Formula presented.). In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several combinatorial criteria of independent interest for detecting various geometric objects on a half-translation surface.",
author = "Valentina Disarlo and Anja Randecker and Robert Tang",
note = "Funding Information: The first and third author are grateful to Mark Bell for many helpful discussions during the Redbud Topology Conference at the University of Oklahoma in April 2016, which led to the formulation of the main question of this paper, as well as the classification of links in Section 4.2 . The first substantial progress on this project was carried out while the authors were in‐residence during the Fall 2016 semester program on {\textquoteleft}Geometric group theory{\textquoteright} at the Mathematical Sciences Research Institute, Berkeley. The authors were supported by the National Science Foundation under Grant Number: DMS 1440140, administered by the Mathematical Sciences Research Institute, during their research stay. Subsequent work was carried out at the Shanks Conference on {\textquoteleft}Low‐dimensional topology and geometry{\textquoteright} at Vanderbilt University in May 2017, at Heidelberg University in December 2017, and at the {\textquoteleft}Geometry of Teichm{\"u}ller space and mapping class groups{\textquoteright} workshop at the University of Warwick in April 2018. The authors thank these venues for their kind hospitality. The authors acknowledge support from U.S. National Science Foundation, Grant Numbers: DMS 1107452, 1107263, 1107367, {\textquoteleft}RNMS: GEometric structures And Representation varieties{\textquoteright} (the GEAR Network). The first author acknowledges support from the Olympia Morata Habilitation Programme of Universit{\"a}t Heidelberg. She is currently funded by the Priority Program 2026 {\textquoteleft}Geometry at Infinity{\textquoteright} of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) DI 2610/2‐1. She also acknowledges support by the European Research Council under Anna Wienhard's ERC‐Consolidator, Grant Number: 614733 (GEOMETRICSTRUCTURES) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC‐2181/1 ‐ 390900948 (the Heidelberg STRUCTURES Cluster of Excellence). The second author acknowledges support from a DFG Forschungsstipendium and from the National Science Foundation under Grant Number: DMS 1607512. The third author acknowledges support from a JSPS KAKENHI Grant‐in‐Aid for Early‐Career Scientists (Number: 19K14541). The authors also thank Sam Taylor for helpful comments, and Kasra Rafi, Chris Judge, and Matt Bainbridge for helpful conversations and encouragement. Finally, the authors thank the anonymous referee for their thorough comments and suggestions for improving the exposition of the paper. Funding Information: The first and third author are grateful to Mark Bell for many helpful discussions during the Redbud Topology Conference at the University of Oklahoma in April 2016, which led to the formulation of the main question of this paper, as well as the classification of links in Section 4.2. The first substantial progress on this project was carried out while the authors were in-residence during the Fall 2016 semester program on {\textquoteleft}Geometric group theory{\textquoteright} at the Mathematical Sciences Research Institute, Berkeley. The authors were supported by the National Science Foundation under Grant Number: DMS 1440140, administered by the Mathematical Sciences Research Institute, during their research stay. Subsequent work was carried out at the Shanks Conference on {\textquoteleft}Low-dimensional topology and geometry{\textquoteright} at Vanderbilt University in May 2017, at Heidelberg University in December 2017, and at the {\textquoteleft}Geometry of Teichm{\"u}ller space and mapping class groups{\textquoteright} workshop at the University of Warwick in April 2018. The authors thank these venues for their kind hospitality. The authors acknowledge support from U.S. National Science Foundation, Grant Numbers: DMS 1107452, 1107263, 1107367, {\textquoteleft}RNMS: GEometric structures And Representation varieties{\textquoteright} (the GEAR Network). The first author acknowledges support from the Olympia Morata Habilitation Programme of Universit{\"a}t Heidelberg. She is currently funded by the Priority Program 2026 {\textquoteleft}Geometry at Infinity{\textquoteright} of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) DI 2610/2-1. She also acknowledges support by the European Research Council under Anna Wienhard's ERC-Consolidator, Grant Number: 614733 (GEOMETRICSTRUCTURES) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence). The second author acknowledges support from a DFG Forschungsstipendium and from the National Science Foundation under Grant Number: DMS 1607512. The third author acknowledges support from a JSPS KAKENHI Grant-in-Aid for Early-Career Scientists (Number: 19K14541). The authors also thank Sam Taylor for helpful comments, and Kasra Rafi, Chris Judge, and Matt Bainbridge for helpful conversations and encouragement. Finally, the authors thank the anonymous referee for their thorough comments and suggestions for improving the exposition of the paper. Publisher Copyright: {\textcopyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",
year = "2022",
month = sep,
doi = "10.1112/topo.12242",
language = "English",
volume = "15",
pages = "1248--1310",
journal = "Journal of Topology",
issn = "1753-8416",
publisher = "John Wiley & Sons, Inc.",
number = "3",
}