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Pullback invariants of Thurston maps


Authors: Sarah Koch, Kevin M. Pilgrim and Nikita Selinger
Journal: Trans. Amer. Math. Soc. 368 (2016), 4621-4655
MSC (2010): Primary 30F60, 32G15, 37F20
DOI: https://doi.org/10.1090/tran/6482
Published electronically: September 24, 2015
MathSciNet review: 3456156
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Abstract: Associated to a Thurston map $ f: S^2 \to S^2$ with postcritical set $ P$ are several different invariants obtained via pullback: a relation $ \mathcal {S}_{P} {\stackrel {f}{\longleftarrow }} \mathcal {S}_{P}$ on the set $ \mathcal {S}_{P}$ of free homotopy classes of curves in $ S^2\setminus P$, a linear operator $ \lambda _f: \mathbb{R}[\mathcal {S}_{P}]\to \mathbb{R}[\mathcal {S}_{P}]$ on the free $ \mathbb{R}$-module generated by $ \mathcal {S}_{P}$, a virtual endomorphism $ \phi _f: \mathrm {PMod}(S^2, P) \dashrightarrow \mathrm {PMod}(S^2, P)$ on the pure mapping class group, an analytic self-map $ \sigma _f: \mathcal {T}(S^2, P) \to \mathcal {T}(S^2, P)$ of an associated Teichmüller space, and an analytic self-correspondence $ X\circ Y^{-1}: \mathcal {M}(S^2, P) \rightrightarrows \mathcal {M}(S^2, P)$ of an associated moduli space. Viewing these associated maps as invariants of $ f$, we investigate relationships between their properties.


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Additional Information

Sarah Koch
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109

Kevin M. Pilgrim
Affiliation: Department of Mathematics, Indiana University, 831 E. Third Street, Bloomington, Indiana 47405

Nikita Selinger
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660

DOI: https://doi.org/10.1090/tran/6482
Received by editor(s): April 2, 2013
Received by editor(s) in revised form: May 10, 2014
Published electronically: September 24, 2015
Article copyright: © Copyright 2015 American Mathematical Society