[Paper Review] The Asymptotic Dimension of Box Spaces for Elementary Amenable Groups

Martin Finn-Sell, Jianchao Wu|arXiv (Cornell University)|Aug 20, 2015

Advanced Operator Algebra Research9 references3 citations

TL;DR

This paper establishes that the asymptotic dimension of box spaces for elementary amenable residually finite groups is bounded above by their Hirsch length, with equality holding for a large class of groups including virtually polycyclic groups and their wreath products with finite abelian groups. The authors develop an inductive method based on group extensions and semi-conjugacy-separating families to prove subadditivity of asymptotic dimension under group extensions, yielding sharp bounds for box spaces of such groups.

ABSTRACT

We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subclass of groups including all virtually polycyclic groups.

Motivation & Objective

• To establish an upper bound for the asymptotic dimension of box spaces of elementary amenable residually finite groups.
• To prove that this bound is sharp for a large class of groups, including virtually polycyclic and wreath products with finite abelian groups.
• To develop a general inductive method for estimating asymptotic dimension of box spaces using group extensions.
• To generalize the coarse geometric treatment of box spaces beyond normal subgroups by introducing the semi-conjugacy-separating condition.

Proposed method

• The authors introduce the concept of semi-conjugacy-separating families of finite-index subgroups, which ensures well-behaved coarse structures on box spaces.
• They prove a subadditivity property: asymptotic dimension of the box space of an extension group is bounded by the sum of the asymptotic dimensions of the kernel and quotient groups.
• The method relies on constructing coarse fibrations and using the coarse geometry of quotient maps to control the asymptotic dimension of fibers.
• Transfinite induction is applied over the Hirsch length filtration of elementary amenable groups to inductively bound the asymptotic dimension of box spaces.
• The proof uses a metric decomposition technique involving weighted metrics on disjoint unions of finite quotients.
• The key technical tool is Lemma 4.1 and Theorem 2.5, which allow control over coarse fibers and enable the subadditivity estimate.

Experimental results

Research questions

• RQ1 What is the asymptotic dimension of the box space of an elementary amenable residually finite group?
• RQ2 How does the asymptotic dimension of a box space behave under group extensions?
• RQ3 Is the Hirsch length a sharp upper bound for the asymptotic dimension of box spaces of elementary amenable groups?
• RQ4 For which classes of groups does the asymptotic dimension of the box space equal the Hirsch length?
• RQ5 Can the coarse structure of box spaces be controlled without restricting to normal subgroups?

Key findings

• The asymptotic dimension of the full box space $\square_{\mathrm{F}}G$ of any $G \in \mathbf{Elem_{RF}}$ is bounded above by the Hirsch length $h(G)$.
• For virtually polycyclic groups and their wreath products with finite abelian groups, the asymptotic dimension of the box space equals the Hirsch length.
• The asymptotic dimension of box spaces behaves subadditively under group extensions: $\mathrm{as\dim}(\square_{\mathrm{F}}G) \leq \mathrm{as\dim}(\square_{\mathrm{F}}N) + \mathrm{as\dim}(\square_{\mathrm{F}}K)$ for $1 \to N \to G \to K \to 1$.
• The same subadditivity holds for the full normal box space $\square_{\mathrm{f}}G$, with the same bound.
• The bound $\mathrm{as\dim}(\square_{\sigma}G) \leq h(G)$ holds for any semi-conjugacy-separating family $\sigma$ of finite-index subgroups.
• For $G = F \wr H$ with $F$ finite abelian and $H$ virtually polycyclic, $\mathrm{as\dim}(\square_{\sigma}G) = h(G) = h(H)$.