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Braiding groups of automorphisms and almost-automorphisms of trees

Rachel Skipper, Matthew C. B. Zaremsky

Abstract

We introduce "braided" versions of self-similar groups and Röver--Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call "self-identical". In particular we use a braided version of the Grigorchuk group to construct a new group called the braided Röver group, which we prove is of type $F_\infty$. Our techniques involve using so called $d$-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

Braiding groups of automorphisms and almost-automorphisms of trees

Abstract

We introduce "braided" versions of self-similar groups and Röver--Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call "self-identical". In particular we use a braided version of the Grigorchuk group to construct a new group called the braided Röver group, which we prove is of type . Our techniques involve using so called -ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties.

Paper Structure

This paper contains 14 sections, 28 theorems, 75 equations, 8 figures.

Table of Contents

  1. Braiding groups of automorphisms of trees
  2. Braiding groups of automorphisms
  3. The braided Grigorchuk group
  4. Braiding groups of almost-automorphisms of trees
  5. Cloning systems
  6. Almost-automorphisms
  7. Braiding groups of almost-automorphisms
  8. Finiteness properties
  9. Stein--Farley complexes
  10. Descending links
  11. Descending links in the braided case
  12. Finiteness properties of the braided Röver group
  13. Polysimplicial structure
  14. Descending links

Key Result

Lemma \oldthetheorem

The kernel of $\pi\colon B_d\wr_X^\infty B_d\to S_d\wr_X^\infty S_d$ is $PB_d\wr_X^\infty PB_d$, which is a direct product of infinitely many copies of $PB_d$.

Figures (8)

  • Figure 1: The braid $\zeta$.
  • Figure 2: An example of $2$-ary cloning on $\operatorname{br}\!W_2$. Here $f\in\operatorname{br}\!\mathop{\mathrm{Aut}}\nolimits(\mathcal{T}_2)$ satisfies the braided wreath recursion $f=\zeta(f,f)$. The picture shows that $(\zeta(\mathop{\mathrm{id}}\nolimits,f))\kappa_2^2 = (\zeta)\vartheta_2^2\phi^{(2)}(f)(\mathop{\mathrm{id}}\nolimits,f,f)$. We use thick lines to indicate the strand getting cloned and the resulting strands.
  • Figure 3: An example of the last step of the verification of (C1) in the proof of Proposition \ref{['prop:brW_n_cloning_system']}. Here $d=3$, $k=3$, and $n=5$.
  • Figure 4: An element of the braided Röver group. Here we draw a triple $(T_-,\beta(f_1,\dots,f_m),T_+)$ with $T_+$ upside down so that $\beta$ is a braid from the leaves of $T_-$ to the leaves of $T_+$, and the $f_i$ label the strands. The element equals $[\wedge,(a,b),\wedge]$, for $\wedge$ the tree with one caret. As indicated in the picture, after expansions this is the same element as $[T,\beta(1,1,a,c),T]$ for $T$ the result of adding a caret to each leaf of $\wedge$ and $\beta\in B_4$ the braid crossing the first strand over the second.
  • Figure 5: An example of the bijective correspondence between elementary $3$-ary forests with $9$ leaves and simplices of ${\mathcal{M}}_3(L_{8})$.
  • ...and 3 more figures

Theorems & Definitions (93)

  • Definition \oldthetheorem: Automorphism
  • Definition \oldthetheorem: Self-similar
  • Definition \oldthetheorem: Self-identical
  • Example \oldthetheorem: Subgroups of the symmetric group
  • Example \oldthetheorem: Grigorchuk group
  • Definition \oldthetheorem: Braided $\mathop{\mathrm{Aut}}\nolimits(\mathcal{T}_d)$
  • Definition \oldthetheorem: Braided self-similar
  • Definition \oldthetheorem: Braided self-identical
  • Remark \oldthetheorem
  • Example \oldthetheorem: Subgroups of the braid group
  • ...and 83 more