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Markovian properties of continuous group actions: Algebraic actions, entropy and the homoclinic group
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null (Ed.)Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.more » « less
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We address the following natural extension problem for group actions: Given a group [Formula: see text], a subgroup [Formula: see text], and an action of [Formula: see text] on a metric space, when is it possible to extend it to an action of the whole group [Formula: see text] on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of [Formula: see text]? We begin by formalizing this problem and present a construction of an induced action which behaves well when [Formula: see text] is hyperbolically embedded in [Formula: see text]. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups.more » « less
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We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfiniteness of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.more » « less
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Abstract Let $$X$$ be a measure space with a measure-preserving action $$(g,x) \mapsto g \cdot x$$ of an abelian group $$G$$. We consider the problem of understanding the structure of measurable tilings $$F \odot A = X$$ of $$X$$ by a measurable tile $$A \subset X$$ translated by a finite set $$F \subset G$$ of shifts, thus the translates $$f \cdot A$$, $$f \in F$$ partition $$X$$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $$F \odot A = X$$ implies $$F^{r} \odot A = X$$ for a large family of integer dilations $$r$$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus $${\mathbb{T}}^{d}$$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.aim.2022.108196
