Regular finite decomposition complexity
Joint with Daniel Kasprowski and Andrew Nicas.
Theorem 1 If has regular FDC, a finite dimensional classifying space and a global upper bound on the orders of its finite subgroups, the assembly map
is a split injection, for every ring .
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 sets, each of which is a union of uniformly bounded and far apart subsets .
Decomposition complexity extends this initial idea. A metric family is a set of metric spaces. A metric family is bounded if diameters of its spaces are uniformly bounded. A subfamily of is a metric familly all of whose spaces are subspaces of spaces of . A map is a set of maps where each occurs at least once. A map is a coarse map if there exists some function such that for every in , metric expansion is controlled by : .
Collections of metric families. E.g. the collection of bounded metric families.
Say a metric family -decomposes over a collection if , ,
- is covered by sets .
- Each is an -disjoint union of spaces .
- The metric family is in .
If and this holds, then this defines for the whole family (note the uniformity of bounds).
Let () denote the collection of all metric families with (resp. ).
One can iterate the procedure.
Say that strongly decomposes over if it 1-decomposes over . Say that weakly decomposes over if it -decomposes over for some .
Guentner-Tessera-Yu’s finite decomposition complexity (resp. weak finite decomposition complexity) involves the collection (resp. ) defined as the smallest collection of metric families that contain and is stable under strong (resp. weak) decomposition.
2. Permanence properties
Say collection has coarse permanence if when and is a coarse embedding, then .
Say collection has fibering permanence if when and is a coarse map such that for every bounded subfamily in , , then .
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 satisfies it.
Let be a finite group acting isometrically on all spaces of family . Define . , are finite quotient permanent, but possibly not , therefore we introduce regular FDC.
3. Regular FDC
Definition 2 Say family regularly decomposes over if there exists a family and a coarse map such that for every bounded subfamily in , .
Definition 3 Let be the smallest collection of metric families containing and is stable under regular decomposition.
Note that contains .
Theorem 4 satisfies all permanence properties we have encountered.
1. be the smallest collection of metric families containing and satisfying fibering permanence. This shows the power of fibering.
3. All known groups in are also in . 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 . Apply Dranishnikov’s embedding theorem into products of trees. Fibering permanence gives that all finite spaces belong as well.