Nearly Maximal Hausdorff Dimension in Finitely Constrained Groups

This work continues the study of the properties of finitely constrained groups of binary tree automorphisms in terms of their Hausdorff dimension. We prove that there are exactly $2^{2d-3}$ finitely constrained groups of binary tree automorphisms with pattern size $d$ and having Hausdorff dimension $1 - \frac{2}{2^{d-1}}$. As part of this proof, we describe the finite patterns that can define such groups, which leads to the fact that all finitely constrained groups of nearly maximal Hausdorff dimension have additive portraits. Additionally, we give an upper bound, in terms of the pattern size $d$, on the number of topologically finitely generated instances with nearly maximal Hausdorff dimension for a given $d$, by applying corollaries of the criteria of Bondarenko and Samoilovych. We also construct a new family of examples of finitely constrained, topologically finitely generated groups with nearly maximal Hausdorff dimension. We conclude by positing several open questions.