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The mapping class group of a minimal subshift

Volume 163 / 2021

Scott Schmieding, Kitty Yang Colloquium Mathematicum 163 (2021), 233-265 MSC: Primary 37B10; Secondary 54H20. DOI: 10.4064/cm7933-2-2020 Published online: 3 July 2020

Abstract

For a homeomorphism $T \colon X \to X$ of a Cantor set $X$, the mapping class group $\mathcal {M}(T)$ is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension $\Sigma _{T}X$. The group $\mathcal {M}(T)$ can be interpreted as the symmetry group of the system $(X,T)$ with respect to the flow equivalence relation. We study $\mathcal {M}(T)$, focusing on the case when $(X,T)$ is a minimal subshift. We show that when $(X,T)$ is a subshift associated to a substitution, the group $\mathcal {M}(T)$ is an extension of $\mathbb {Z}$ by a finite group; for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of $(X,T)$. When $(X,T)$ is a minimal subshift of linear complexity satisfying a no-infinitesimals condition, we show that $\mathcal {M}(T)$ is virtually abelian. We also show that when $(X,T)$ is minimal, $\mathcal {M}(T)$ embeds into the Picard group of the crossed product algebra $C(X) \rtimes _{T} \mathbb {Z}$.

Authors

  • Scott SchmiedingDepartment of Mathematics
    University of Denver
    Denver, CO 80210, U.S.A.
    e-mail
  • Kitty YangDepartment of Mathematics
    Northwestern University
    Evanston, IL 60208, U.S.A.
    e-mail

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