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Weakly maximal subgroups of branch groups

Paul-Henry Leemann

Abstract

Let $G$ be a branch group acting by automorphisms on a rooted tree $T$. Stabilizers of infinite rays in $T$ are examples of weakly maximal subgroups of $G$ (subgroups that are maximal among subgroups of infinite index), but in general they are not the only examples. In this note we describe two families of weakly maximal subgroups of branch groups. We show that, for the first Grigorchuk group as well as for the torsion GGS groups, every weakly maximal subgroup belongs to one of these families. The first family is a generalization of stabilizers of rays, while the second one consists of weakly maximal subgroups with a block structure. We obtain different equivalent characterizations of these families in terms of finite generation, the existence of a trivial rigid stabilizer, the number of orbit-closures for the action on the boundary of the tree or by the means of sections.

Weakly maximal subgroups of branch groups

Abstract

Let be a branch group acting by automorphisms on a rooted tree . Stabilizers of infinite rays in are examples of weakly maximal subgroups of (subgroups that are maximal among subgroups of infinite index), but in general they are not the only examples. In this note we describe two families of weakly maximal subgroups of branch groups. We show that, for the first Grigorchuk group as well as for the torsion GGS groups, every weakly maximal subgroup belongs to one of these families. The first family is a generalization of stabilizers of rays, while the second one consists of weakly maximal subgroups with a block structure. We obtain different equivalent characterizations of these families in terms of finite generation, the existence of a trivial rigid stabilizer, the number of orbit-closures for the action on the boundary of the tree or by the means of sections.

Paper Structure

This paper contains 16 sections, 81 theorems, 46 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction and statements of the main results
  2. Organization of the paper
  3. Acknowledgment
  4. Definitions and preliminaries
  5. Some examples of branch groups
  6. Generalized parabolic subgroups
  7. Non-rigidity tree
  8. Sections of weakly maximal subgroups
  9. Block subgroups
  10. Level-transitive weakly maximal subgroups
  11. Weakly maximal subgroups of the first Grigorchuk group
  12. More examples of weakly maximal subgroups of the first Grigorchuk group
  13. The subgroup $W_P$
  14. The subgroup $W_T$
  15. Sections of weakly maximal subgroups of the first Grigorchuk group
  16. ...and 1 more sections

Key Result

Proposition 1.1

If $G\leq\mathop{\mathrm{Aut}}\nolimits(T)$ is weakly branch, then all the $\mathop{\mathrm{Stab}}\nolimits_G(\xi)$ for $\xi\in\partial T$ are infinite and pairwise distinct. Moreover, if $G$ is branch, then all these subgroups are weakly maximal.

Figures (4)

  • Figure 1: A diagonal block subgroup over $K$ with supporting vertex set of cardinality $3$.
  • Figure 2: A block subgroup with supporting partition $\{\{000,001\},1\}$. Here $H=H_1\cdot H_2$ with $H_1$ a block subgroup over $K$ with supporting partition $\{000,001\}$ and $H_2$ a block subgroup over $B$ with supporting partition $\{B\}$.
  • Figure 3: The lattice of subgroups of ${\mathcal{G}}$ containing $B$. Each subgroup has index $2$ in the one above it.
  • Figure 4: The stabilizer in ${\mathcal{G}}$ of the vertex $1^n$.

Theorems & Definitions (153)

  • Proposition 1.1: MR1841750MR2893544
  • Theorem 1.2: MR3478865
  • Theorem 1.3: MR3478865
  • Lemma 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Proposition 1.10
  • ...and 143 more