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On finitely generated Hausdorff spectra of groups acting on rooted trees

Jorge Fariña-Asategui, Oihana Garaialde Ocaña, Jone Uria-Albizuri

TL;DR

The paper resolves central questions about finitely generated Hausdorff spectra for groups acting on rooted trees by proving that closures of finitely generated regular branch groups have complete spectra $[0,1]$ with respect to the level-stabilizer filtration, and extends this to level-transitive iterated wreath products $W_H$. It introduces the concept of nice filtrations and develops a framework to realize any Hausdorff dimension via explicit constructions of finitely generated subgroups, including $2$-generated subgroups of $W_p$ with prescribed dimensions. It further generalizes Klopsch’s results to groups with fully dimensional rigid stabilizers, derives spectra for well-studied branch and weakly branch groups, and analyzes standard filtrations for the first Grigorchuk group to obtain full spectra there as well. The results provide explicit, constructive evidence toward conjectures of Abért–Virág and advance understanding of how spectrum behavior persists across broad families of groups acting on rooted trees, with potential implications for $p$-adic analytic pro-$p$ groups and related profinite structures.

Abstract

We answer two longstanding questions of Klopsch (1999) and Shalev (2000) by proving that the finitely generated Hausdorff spectrum of the closure of a finitely generated regular branch group with respect to the level-stabilizer filtration is the full interval [0,1]. Previous work related to the above questions include the celebrated result of Abért and Virág on the completeness of the finitely generated Hausdorff spectrum of the group of $p$-adic automorphisms $W_p$. We extend their result to any level-transitive iterated wreath product acting on a regular rooted tree, and more importantly, provide the first explicit examples of finitely generated subgroups with prescribed Hausdorff dimension. In fact, in $W_p$ such subgroups can be taken to be 2-generated, giving further evidence to a conjecture of Abért and Virág. Furthermore, we extend a well-known result of Klopsch (1999) for branch groups to closed level-transitive groups with fully dimensional rigid stabilizers: these have full Hausdorff spectrum with respect to the level-stabilizer filtration. In turn, we determine, for the first time, the Hausdorff spectum of many well-studied families of weakly branch groups. As an additional new contribution, we define {\it nice filtrations} (which include the level-stabilizer filtration). In fact, the first and the last result can be proven more generally with respect to these filtrations (under possibly some further assumptions). Finally, we also consider two standard filtrations, namely the $2$-central lower series, and the dimension subgroup or Jenning-Zassenhaus series in the closure of the first Grigorchuck group, and show that its finitely generated Hausdorff spectra with respect to these filtrations is the full interval [0,1], answering a recent question of de las Heras and Thillaisundaram (2022). This is obtained via a reduction to properties concerning to nice filtrations.

On finitely generated Hausdorff spectra of groups acting on rooted trees

TL;DR

The paper resolves central questions about finitely generated Hausdorff spectra for groups acting on rooted trees by proving that closures of finitely generated regular branch groups have complete spectra with respect to the level-stabilizer filtration, and extends this to level-transitive iterated wreath products . It introduces the concept of nice filtrations and develops a framework to realize any Hausdorff dimension via explicit constructions of finitely generated subgroups, including -generated subgroups of with prescribed dimensions. It further generalizes Klopsch’s results to groups with fully dimensional rigid stabilizers, derives spectra for well-studied branch and weakly branch groups, and analyzes standard filtrations for the first Grigorchuk group to obtain full spectra there as well. The results provide explicit, constructive evidence toward conjectures of Abért–Virág and advance understanding of how spectrum behavior persists across broad families of groups acting on rooted trees, with potential implications for -adic analytic pro- groups and related profinite structures.

Abstract

We answer two longstanding questions of Klopsch (1999) and Shalev (2000) by proving that the finitely generated Hausdorff spectrum of the closure of a finitely generated regular branch group with respect to the level-stabilizer filtration is the full interval [0,1]. Previous work related to the above questions include the celebrated result of Abért and Virág on the completeness of the finitely generated Hausdorff spectrum of the group of -adic automorphisms . We extend their result to any level-transitive iterated wreath product acting on a regular rooted tree, and more importantly, provide the first explicit examples of finitely generated subgroups with prescribed Hausdorff dimension. In fact, in such subgroups can be taken to be 2-generated, giving further evidence to a conjecture of Abért and Virág. Furthermore, we extend a well-known result of Klopsch (1999) for branch groups to closed level-transitive groups with fully dimensional rigid stabilizers: these have full Hausdorff spectrum with respect to the level-stabilizer filtration. In turn, we determine, for the first time, the Hausdorff spectum of many well-studied families of weakly branch groups. As an additional new contribution, we define {\it nice filtrations} (which include the level-stabilizer filtration). In fact, the first and the last result can be proven more generally with respect to these filtrations (under possibly some further assumptions). Finally, we also consider two standard filtrations, namely the -central lower series, and the dimension subgroup or Jenning-Zassenhaus series in the closure of the first Grigorchuck group, and show that its finitely generated Hausdorff spectra with respect to these filtrations is the full interval [0,1], answering a recent question of de las Heras and Thillaisundaram (2022). This is obtained via a reduction to properties concerning to nice filtrations.

Paper Structure

This paper contains 15 sections, 37 theorems, 143 equations, 1 figure.

Table of Contents

  1. introduction
  2. Preliminaries
  3. The $\mathbf{m}$-adic tree and its group of automorphisms
  4. Subgroups of $\mathrm{Aut}~T_{\mathbf{m}}$
  5. Hausdorff dimension
  6. Nice filtrations and Hausdorff dimension
  7. Definition and main examples of nice filtrations
  8. General results for nice filtrations
  9. Hausdorff spectrum and normal Hausdorff spectrum
  10. Finitely generated branch groups
  11. Iterated wreath products
  12. Generalized Šunić groups
  13. Siegenthaler's construction
  14. 2-generated subgroups of $W_p$ with prescribed Hausdorff dimension
  15. Standard filtrations: the first Grigorchuk group

Key Result

Theorem 1.1

The Hausdorff spectrum of a profinite branch group with respect to the level-stabilizer filtration $\mathcal{S}=\{\mathrm{St}(n)\}_{n\ge 1}$ is the full interval $[0,1]$.

Figures (1)

  • Figure 1: The generators of $L_X$ for $X=\{x_n\}_{n\ge 1}\subseteq T_m$, where $k_j^*$ represents the appropriate conjugate of $k_j$ appearing in the proof of \ref{['lemma: construction of LI']}.

Theorems & Definitions (65)

  • Theorem 1.1: see KlopschPhD
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • ...and 55 more