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FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION

Part of: Topological dynamics Compact groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  11 November 2015

ANDREW PENLAND  and
ZORAN ŠUNIĆ
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]
*
✉[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 $d$.

Keywords

finitely constrained groupsgroups acting on treessymbolic dynamics on treesgroup tree shiftsHausdorff dimension

MSC classification

Primary: 20E08: Groups acting on trees
Secondary: 22C05: Compact groups 37B10: Symbolic dynamics 37B99: None of the above, but in this section
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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