The transfinite decomposition rank of the direct sum of iterated wreath products of the integers

Guentner, Tessera and Yu organized metric families into a transfinite hierarchy of classes $\mathfrak{D}_\alpha$ , indexed by countable ordinals, which underlies finite decomposition complexity (FDC), a coarse-geometric property that implies Yu's property A and supports their proof of the stable Borel conjecture for aspherical manifolds with FDC fundamental group. To witness that the hierarchy is genuinely transfinite, they proved that the group $G=\bigoplus_{n\ge 1}G_n$ , a direct sum of $n$ -fold iterated wreath products of $\mathbb{Z}$ , lies in $\mathfrak{D}_{\omega^2}$ , and posed as an open question whether $G$ already lies in $\mathfrak{D}_\alpha$ for some ordinal $\alpha<\omega^2$ . The question remains open.

Background

Asymptotic dimension (asdim), introduced by Gromov, is a large-scale analogue of the Lebesgue covering dimension for metric spaces and, in particular, for finitely generated groups equipped with a word metric: a metric space has asdim at most $n$ if for every scale $r>0$ it can be covered by $n+1$ families of uniformly bounded sets, each family being $r$ -disjoint (any two distinct members of the same family lie more than $r$ apart). Many important groups have infinite asymptotic dimension, and Guentner, Tessera and Yu proposed finite decomposition complexity (FDC) as a powerful generalization. Their framework works with metric families, that is, countable collections of metric spaces, a single metric space being viewed as the one-element family containing it: a family $\mathcal{X}$ is $r$ -decomposable over a family $\mathcal{Y}$ if every space $X$ in $\mathcal{X}$ splits as $X_0\cup X_1$ , where each $X_i$ is an $r$ -disjoint union of spaces from $\mathcal{Y}$ , and $\mathcal{X}$ is decomposable over a class $\mathcal{C}$ of families if for every scale $r>0$ it is $r$ -decomposable over some family in $\mathcal{C}$ . Starting from the class of bounded families (those with a uniform bound on diameters) and iterating the decomposability operation through the ordinals yields classes $\mathfrak{D}_\alpha$ : $\mathfrak{D}_0$ is the class of bounded families, and for $\alpha>0$ , $\mathfrak{D}_\alpha$ consists of the families decomposable over $\bigcup_{\beta<\alpha}\mathfrak{D}_\beta$ . A metric space, and hence a group, is placed in $\mathfrak{D}_\alpha$ when it belongs to $\mathfrak{D}_\alpha$ as a one-element family; countable groups are equipped with proper left-invariant metrics (every countable group admits one, and any two such metrics are coarsely equivalent). A group has FDC if and only if it lies in $\mathfrak{D}_\alpha$ for some countable ordinal $\alpha$ . FDC implies property A, a weak amenability-type regularity property of Yu that yields coarse embeddability into Hilbert space, and it suffices for the stable Borel conjecture for aspherical manifolds whose fundamental group has FDC (the Borel conjecture asserts that a closed aspherical manifold is determined up to homeomorphism by its homotopy type). The $\mathfrak{D}$ -hierarchy is graded by familiar examples: $\mathbb{Z}^n$ has asdim $n$ and lies in $\mathfrak{D}_n$ ; the direct sum $\bigoplus\mathbb{Z}$ of countably many copies of $\mathbb{Z}$ has infinite asdim and first appears at the first limit level $\mathfrak{D}_\omega$ ; and the restricted wreath product $\mathbb{Z}\wr\mathbb{Z}=\bigoplus_{\mathbb{Z}}\mathbb{Z}\rtimes\mathbb{Z}$ , the semidirect product of the direct sum of copies of $\mathbb{Z}$ indexed by $\mathbb{Z}$ under the shift action of $\mathbb{Z}$ , lies in $\mathfrak{D}_{\omega+1}$ . Writing $G_n$ for the $n$ -fold iterated wreath product $\cdots((\mathbb{Z}\wr\mathbb{Z})\wr\mathbb{Z})\cdots$ of $n$ copies of $\mathbb{Z}$ , Guentner–Tessera–Yu proved that the direct sum $G=\bigoplus_{n\ge 1}G_n$ lies in $\mathfrak{D}_{\omega^2}$ (the level obtained by taking suprema over the increasing finite-iteration levels), and they asked whether some ordinal below $\omega^2$ already suffices. One should distinguish this hierarchy from Radul's transfinite asymptotic dimension, a different transfinite extension of asdim defined directly through dimension-style covering functions, under which the wreath product satisfies $\operatorname{trasdim}(\mathbb{Z}\wr\mathbb{Z})\le\omega+1$ ; that bound does not transfer to the $\mathfrak{D}$ -classes. By contrast, in the decomposition hierarchy itself the iterated wreath product $(\mathbb{Z}\wr\mathbb{Z})\wr\mathbb{Z}$ lies in $\mathfrak{D}_{\omega+1}$ ; the exact $\mathfrak{D}$ -rank of $G$ remains the open question.

Problem Statement

Let $G_n$ denote the $n$ -fold iterated restricted wreath product of $n$ copies of $\mathbb{Z}$ , that is $G_n=\cdots((\mathbb{Z}\wr\mathbb{Z})\wr\mathbb{Z})\cdots\wr\mathbb{Z}$ , and let $G=\bigoplus_{n\ge 1}G_n$ be their direct sum. The group $G$ is countable but not finitely generated; equip it with a proper left-invariant metric, the standing convention of Guentner, Tessera and Yu for countable groups (any two such metrics are coarsely equivalent). Guentner, Tessera and Yu proved that $G\in\mathfrak{D}_{\omega^2}$ , where $\mathfrak{D}_\alpha$ is their transfinite hierarchy of decomposition-complexity classes of metric families ( $\mathfrak{D}_0$ the bounded families, $\mathfrak{D}_\alpha$ the families decomposable over lower classes; a group belongs to $\mathfrak{D}_\alpha$ when, viewed as a one-element metric family, it lies in that class). Determine whether there exists a countable ordinal $\alpha<\omega^2$ such that $G\in\mathfrak{D}_\alpha$ ; since $G\in\mathfrak{D}_{\omega^2}$ is known and the hierarchy is cumulative, this asks whether the least ordinal $\alpha$ with $G\in\mathfrak{D}_\alpha$ is strictly below $\omega^2$ or equals $\omega^2$ .

References

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