## Notes of David Rosenthal’s Southampton lecture 29-03-2017

Regular finite decomposition complexity

Joint with Daniel Kasprowski and Andrew Nicas.

Theorem 1 If ${G}$ has regular FDC, a finite dimensional classifying space and a global upper bound on the orders of its finite subgroups, the assembly map

$\displaystyle \begin{array}{rcl} H_*^G(EG;\mathbb{K}_R)\rightarrow K_*(R[G]) \end{array}$

is a split injection, for every ring ${R}$.

This a generalization of Guentner-Tessera-Yu’s theorem, with a similar proof, but handier technical details we allow to hope for further generalizations.

1. From asymptotic dimension to decomposition complexity

Asymptotic dimension requires a decomposition of the space in ${n+1}$ sets, each of which is a union of uniformly bounded and far apart subsets ${X_{ij}}$.

Decomposition complexity extends this initial idea. A metric family ${\mathcal{X}}$ is a set of metric spaces. A metric family is bounded if diameters of its spaces are uniformly bounded. A subfamily ${\mathcal{Y}}$ of ${\mathcal{X}}$ is a metric familly all of whose spaces are subspaces of spaces of ${\mathcal{X}}$. A map ${F:\mathcal{X}\rightarrow\mathcal{Y}}$ is a set of maps ${f:X\rightarrow Y}$ where each ${X\in\mathcal{X}}$ occurs at least once. A map ${F:\mathcal{X}\rightarrow\mathcal{Y}}$ is a coarse map if there exists some function ${\rho}$ such that for every ${f:X\rightarrow Y}$ in ${F}$, metric expansion is controlled by ${\rho}$: ${d(f(x),f(x'))\leq \rho(d(x,x'))}$.

Collections of metric families. E.g. the collection ${\mathfrak{B}}$ of bounded metric families.

Say a metric family ${\mathcal{X}}$ ${n}$-decomposes over a collection ${\mathfrak{C}}$ if ${\forall R>0}$, ${\forall X\in \mathcal{X}}$,

1. ${X}$ is covered by ${n+1}$ sets ${X_i}$.
2. Each ${X_i}$ is an ${R}$-disjoint union of spaces ${X_{ij}}$.
3. The metric family ${\{X_{ij}\}}$ is in ${\mathfrak{C}}$.

If ${\mathfrak{C}=\mathfrak{B}}$ and this holds, then this defines ${asdim\leq n}$ for the whole family ${\mathcal{X}}$ (note the uniformity of bounds).

Let ${\mathfrak{A}}$ (${\mathfrak{A}_n}$) denote the collection of all metric families with ${asdim<\infty}$ (resp. ${asdim\leq n}$).

One can iterate the procedure.

Say that ${\mathcal{X}}$ strongly decomposes over ${\mathfrak{C}}$ if it 1-decomposes over ${\mathfrak{C}}$. Say that ${\mathcal{X}}$ weakly decomposes over ${\mathfrak{C}}$ if it ${n}$-decomposes over ${\mathfrak{C}}$ for some ${n}$.

Guentner-Tessera-Yu’s finite decomposition complexity (resp. weak finite decomposition complexity) involves the collection ${\mathfrak{D}}$ (resp. ${w\mathfrak{D}}$) defined as the smallest collection of metric families that contain ${\mathfrak{B}}$ and is stable under strong (resp. weak) decomposition.

2. Permanence properties

Say collection ${\mathfrak{C}}$ has coarse permanence if when ${\mathcal{Y}\in\mathfrak{C}}$ and ${F:\mathcal{X}\rightarrow \mathcal{Y}}$ is a coarse embedding, then ${\mathcal{X}\in\mathfrak{C}}$.

Say collection ${\mathfrak{C}}$ has fibering permanence if when ${\mathcal{Y}\in\mathfrak{C}}$ and ${F:\mathcal{X}\rightarrow \mathcal{Y}}$ is a coarse map such that for every bounded subfamily ${\mathcal{U}}$ in ${\mathcal{Y}}$, ${F^{-1}(\mathcal{U})\in\mathfrak{C}}$, then ${\mathcal{X}\in\mathfrak{C}}$.

Simultaneously, one defines finite amalgamation, finite union, union, limit permanence. These play a role in establishing permanence properties of groups, such as free products, direct unions, and extensions.

Finite quotient permanence is motivating our work. It is unknown wether Guentner-Tessera-Yu’s collection ${\mathfrak{D}}$ satisfies it.

Let ${F}$ be a finite group acting isometrically on all spaces of family ${\mathcal{X}}$. Define ${F\setminus \mathcal{X}=\{F\setminus X\,;\,X\in\mathcal{X}\}}$. ${\mathfrak{A}}$, ${w\mathfrak{D}}$ are finite quotient permanent, but possibly not ${\mathfrak{D}}$, therefore we introduce regular FDC.

3. Regular FDC

Definition 2 Say family ${\mathcal{X}}$ regularly decomposes over ${\mathfrak{C}}$ if there exists a family ${\mathcal{Y}\in\mathfrak{A}}$ and a coarse map ${F:\mathcal{X}\rightarrow\mathcal{Y}}$ such that for every bounded subfamily ${\mathcal{U}}$ in ${\mathcal{Y}}$, ${F^{-1}(\mathcal{U})\in\mathfrak{C}}$.

Definition 3 Let ${\mathfrak{R}}$ be the smallest collection of metric families containing ${\mathfrak{B}}$ and is stable under regular decomposition.

Note that ${\mathfrak{R}}$ contains ${\mathfrak{A}}$.

Theorem 4 ${\mathfrak{R}}$ satisfies all permanence properties we have encountered.

3.1. Corollaries

1. ${\mathfrak{R}}$ be the smallest collection of metric families containing ${\mathfrak{A}}$ and satisfying fibering permanence. This shows the power of fibering.

2. ${\mathfrak{R}\subset\mathfrak{D}}$.

3. All known groups in ${\mathfrak{D}}$ are also in ${\mathfrak{R}}$. Indeed, proof in each case relies on permanence properties only.

4. Ramsey-Ramras: regular FDC is extendable to relatively hyperbolic groups (peripheral subgroups have it implies that group has it).

Permanence properties, especially fibering permanence, are powerful. Here is a typical argument. Show trees belong to ${\mathfrak{R}}$. Apply Dranishnikov’s embedding theorem into products of trees. Fibering permanence gives that all finite ${asdim}$ spaces belong as well.