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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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