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FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION

Published online by Cambridge University Press:  11 November 2015

ANDREW PENLAND*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA email [email protected]
ZORAN ŠUNIĆ
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA email [email protected]
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Abstract

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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 $d$.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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