Finitely Constrained Groups

Abstract

This work investigates three aspects of the theory of finitely constrained groups, motivated by questions first asked by Rostislav Grigorchuk when he introduced the subject in 2005. The first topic is Hausdorff dimension of finitely constrained groups of p-adic tree automorphisms. The set of possible values of Hausdorff dimension for such a group is known, and we are able to show that every value in this set actually occurs. The second topic, related to the first, is topological finite generation of finitely constrained groups of p-adic tree automorphisms. Relatively little is known about which values of Hausdorff dimension occur for topologically finitely generated, finitely constrained groups of p-adic tree automorphisms. We are able to show that certain values can not occur as the Hausdorff dimension a topologically finitely generated, finitely constrained group of p-adic automorphisms defined by patterns of size d. We discuss finitely constrained groups of binary tree automorphisms with pattern size d ≥ 5 and Hausdorff dimension 1 – 2/2^d-1 ; the issue of topological finite generation for these groups is more challenging. We provide explicit constructions of new examples of finitely constrained groups and calculate their Hausdorff dimension. Finally, we study the portraits of self-similar groups using well-known ideas from the theory of tree automata, with particular focus on examples which separate certain classes of tree languages. These self-similar groups generalize the usual notion of self-similar groups, and we show that some well-known results extend to this more general case. From the tree language perspective, self-similar groups whose portraits form sofic tree shifts are of particular interest. We conclude by posing many questions for future study.