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FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION
Published online by Cambridge University Press: 11 November 2015
Abstract
We prove that if $G_{P}$ is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group $P$ of pattern size $d$, $d\geq 2$, and if $G_{P}$ has maximal Hausdorff dimension (equal to $1-1/2^{d-1}$), then $G_{P}$ is not topologically finitely generated. We describe precisely all essential pattern groups $P$ that yield finitely constrained groups with maximal Hausdorff dimension. For a given size $d$, $d\geq 2$, there are exactly $2^{d-1}$ such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth $d$.
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MSC classification
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 108 - 123
- Copyright
- © 2015 Australian Mathematical Publishing Association Inc.
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