FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION
Andrew Penland, Zoran Šunić · Journal of the Australian Mathematical Society · 2015
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 $d$ .