Finitely constrained groups of maximal Hausdorff dimension

TL;DR

This paper characterizes finitely constrained groups of binary rooted tree automorphisms with maximal Hausdorff dimension, showing they are not topologically finitely generated and precisely describing the pattern groups that produce such maximal dimension.

Contribution

It provides a complete classification of essential pattern groups that yield finitely constrained groups with maximal Hausdorff dimension, including their count and maximality properties.

Findings

Finitely constrained groups with maximal Hausdorff dimension are not topologically finitely generated.

Exactly 2^{d-1} such pattern groups exist for each pattern size d > 1.

All these pattern groups are maximal in the automorphism group of the finite rooted tree.

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>1, 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 Haudorff dimension. For a given size d, d>1, 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.