Finitely Constrained Groups of Maximal Hausdorff Dimension

Abstract

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 ≥ 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 Haudorff dimension. For a given size d, d ≥ 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.

Nearly Maximal Hausdorff Dimension in Finitely Constrained Groups

“…In this paper, we consider finitely constrained groups defined by patterns of size d and having Hausdorff dimension 1 − 2 2 d−1 . This particular value is interesting because it is the largest possible for a topologically finitely generated, finitely constrained group of binary tree automorphisms with pattern size d.This work may be viewed as a continuation of a previous joint work with Zoran Šunić [33], which considered these properties for finitely constrained groups with pattern size d and Hausdorff dimension 1 − 1 2 d−1 . We call the value 1 − 2 2 d−1 nearly maximal Hausdorff dimension for a finitely constrained group defined by patterns of size d, since 1 − 1 2 d−1 is the largest possible value for such a group.…”

“…The Hausdorff dimension of these groups was calculated by Pink [34]. The fact that these groups are finitely constrained, and the defining patterns some of the groups, is discussed in [33,Section 5]. Recently, Samoilovych [36] has independently given a description of the defining patterns for the topological closures of self-similar iterated monodromy groups of post-critically finite polynomials.…”

“…It is known that a finitely constrained group can only have Hausdorff dimension equal to 1 if the group is all of Aut(X * ), so the value 1 − In a previous joint work [33], the current author and Zoran Šunić studied the finitely constrained groups of binary tree automorphisms with maximal Hausdorff dimension, proving that a group with these properties can not be topologically finitely generated. An explicit description for the essential pattern groups of size d that define such groups was also given (see Theorem 5 in this work).…”

“…For finitely constrained groups with nearly maximal Hausdorff dimension, already the question of topological finite generation is more subtle, with known examples and counterexamples. As noted in [33], the constructions of Bartholdi and Nekrashevych in [12] give, for each d ≥ 5, exactly d − 2 distinct examples of topologically finitely generated, finitely constrained groups defined by patterns of size d and having nearly maximal Hausdorff dimension.…”

A language hierarchy and Kitchens-type theorem for self-similar groups

“…Bondarenko and Samoilovych provided criteria to establish that a finitely constrained group is topologically finitely generated [7]. In [23], the present authors applied these criteria to show that finitely constrained groups having a certain Hausdorff dimension were not topologically finitely generated. Additionally, the results of Fernández-Alcober and Zugadi-Reizabal in [12] show that if G is a GGS-group acting on a p-regular tree where p is an odd prime, and if G has a non-constant defining vector, then G is finitely constrained (this follows from combining the results of Lemma 3.4 of [12] and Theorem 1).…”