The Farrell-Jones Conjecture and (coarse) flow spaces, II
1. Coarse flow spaces and N--amenability
Let be a collection of subgroups of . Let act on space . Say that is an -set if if translates of are either disjoint or coincide, and there exists such that iff .
Proposition 1 (Guentner-Willett-Yu) The action of on is N--amenable iff for all finite subsets , there exists an -cover of of dimension , and which is -wide, meaning that , is contained in a piece of the cover (“-wide”).
1.1. Hyperbolic groups
Theorem 2 (Bartels-Lueck-Reich) If is hyperbolic, the action of on is N-VCyc-amenable.
The proof relies on the construction of a coarse geodesic flow. Define
where lies at distance of some geodesic from to . Define the subset . This is thought of as if there indeed was a flow .
We first produce a cover of , where the required Lebesgue number is (“long and thin cover”). The key points are
- Coarse flow lines are quasi-isometric to lines, hence they are doubling.
- -action on is proper.
- The isotropy for pairs in is in Vcyc.
Then we use the map mapping to pushing a distance in the direction of along some geodesic, to pull-back the cover. For every finite set , the obtained cover is -wide, for and large enough.
1.3. Relative hyperbolic groups
We take Bowditch’s definition. Let act on a graph which is hyperbolic and fine (for any , finitely many cycles of length at most through any edge), with finite edge stabilizers. Vertex stabilizers need not be finite, we call them parabolic subgroups. Let where is the set of vertices with infinite valency. In the observer topology, this is a compact Hausdorff space.
Theorem 3 (Bartels) The action of on is N-(VCyc )-amenable, where is the family of parabolic subgroups.
On the set of edges of , there exists a proper -invariant metric, called angle metric: angle
Large angles force geodesic triangles to degenerate: if angle of and from is large, belongs to one of the sides , for any .
Fix a base point and a large . Let angle from of and is . A covering of this part is produced as in the hyperbolic case.
For a vertex of infinite valency, and a large . Let be the set of viewed from with angle , and be the collection of all This may be infinite, but it turns out that only the first point of along a ray matters. This is not a coarse notion. To fix this, one needs introduce a second scale . In this way, adding -sets to covers of ‘s, one manages to construct an -wide cover .
1.4. Mapping class groups
Let act on Teichmuller space . Use Thurston’s compactification. Let be the class of subgroups of that fix virtually a point of .
Theorem 4 (Bartels-Bestvina) The action of on is N-(Vcyc )-amenable.
To prove this, we we fix and cocompact part of and we split into the part where some Teichmuller geodesic from to is entirely contained in , and its complement. For this part, use
Masur: Teichmuller rays in converge in .
Minsky: Teichmuller geodesics in
hence hyperbolic arguments apply.
For the remainder, there is an analogue of vertices of infinite degree, subsurfaces of . There is an analogue of angles, the Masur-Minsky subsurface projection and distances measured in the curve complex. The projection complex of Bestvina-Bromberg-Fujiwara (which they used to estimate the asymptotic dimension of ) plays the role of the hyperbolic graph .