Notes of Arthur Bartels’ second Southampton lecture 29-03-2017

The Farrell-Jones Conjecture and (coarse) flow spaces, II

1. Coarse flow spaces and N-{\mathcal{F}}-amenability

Let {\mathcal{F}} be a collection of subgroups of {G}. Let {G} act on space {Z}. Say that {U\subset Z} is an {\mathcal{F}}-set if if translates of {U} are either disjoint or coincide, and there exists {F\in\mathcal{F}} such that {gU=U} iff {u\in F}.

Proposition 1 (Guentner-Willett-Yu) The action of {G} on {X} is N-{\mathcal{F}}-amenable iff for all finite subsets {E\subset G}, there exists an {\mathcal{F}}-cover of {G\times X} of dimension {\leq N}, and which is {E}-wide, meaning that {\forall (g,x)\in G\times X}, {gE\times\{x\}} is contained in a piece of the cover (“{E}-wide”).

1.1. Hyperbolic groups

Theorem 2 (Bartels-Lueck-Reich) If {G} is hyperbolic, the action of {G} on {\partial G} is N-VCyc-amenable.

1.2. Proof

The proof relies on the construction of a coarse geodesic flow. Define

\displaystyle  \begin{array}{rcl}  CF=\{(v_-,v_0,\xi_+)\in (G\cup\partial G)\times G\times\partial G\} \end{array}

where {v_0} lies at distance {\leq\delta} of some geodesic from {v_-} to {\xi_+}. Define the subset {\Phi_R(v_-,v_0,\xi)=(\{v_-\}\times B_R(v_0)\times \{xi_+\})\cap CF}. This is thought of as {\Phi_{[-R,R]}(v_-,v_0,\xi)} if there indeed was a flow {(\Phi_t)}.

We first produce a cover of {CF}, where the required Lebesgue number is {R} (“long and thin cover”). The key points are

  • Coarse flow lines are quasi-isometric to lines, hence they are doubling.
  • {G}-action on {G} is proper.
  • The isotropy for pairs in {G\cup\partial G\times\partial G} is in Vcyc.

Then we use the map {\iota_\tau:G\times \partial G\rightarrow CF} mapping {(g,\xi)} to {(g,v_0,\xi)} pushing {g} a distance {\tau} in the direction of {\xi} along some geodesic, to pull-back the cover. For every finite set {E\subset G}, the obtained cover is {E}-wide, for {R} and {\tau} large enough.

1.3. Relative hyperbolic groups

We take Bowditch’s definition. Let {G} act on a graph {X} which is hyperbolic and fine (for any {R}, finitely many cycles of length at most {R} through any edge), with finite edge stabilizers. Vertex stabilizers need not be finite, we call them parabolic subgroups. Let {\Delta=\partial X\cup V} where {V} is the set of vertices with infinite valency. In the observer topology, this is a compact Hausdorff space.

Theorem 3 (Bartels) The action of {G} on {\Delta} is N-(VCyc {\cup\mathcal{P}})-amenable, where {\mathcal{P}} is the family of parabolic subgroups.

On the set of edges of {X}, there exists a proper {G}-invariant metric, called angle metric: angle

Large angles force geodesic triangles to degenerate: if angle of {x} and {y} from {v} is large, {v} belongs to one of the sides {[x,z]\cup[z,y]}, for any {z}.

Fix a base point {x_0} and a large {\Theta}. Let {(G\times\Delta)_{\leq\Theta}=\{(g,\xi)\,;\,} angle from {v} of {gx_0} and {\xi} is {\leq\Theta\}}. A covering of this part is produced as in the hyperbolic case.

For a vertex {v\in V} of infinite valency, and a large {\eta}. Let {U(v,\eta)} be the set of {(g,\xi)} viewed from {v} with angle {>\eta}, and {U_\eta} be the collection of all {U(v,\eta)} This may be infinite, but it turns out that only the first point of {U_\eta} along a ray {[g,\xi)} matters. This is not a coarse notion. To fix this, one needs introduce a second scale {\eta'}. In this way, adding {\mathcal{P}}-sets to covers of {U_\eta}‘s, one manages to construct an {E}-wide cover .

1.4. Mapping class groups

Let {G=MCG(\Sigma)} act on Teichmuller space {T}. Use Thurston’s compactification. Let {\mathcal{F}} be the class of subgroups of {G} that fix virtually a point of {\bar T}.

Theorem 4 (Bartels-Bestvina) The action of {MCG} on {\partial T} is N-(Vcyc {\cup\mathcal{F}})-amenable.

To prove this, we we fix and cocompact part {K} of {T} and we split {G\times\partial T} into the part where some Teichmuller geodesic from {g} to {\xi} is entirely contained in {K}, and its complement. For this part, use

Masur: Teichmuller rays in {K} converge in {\bar T}.

Minsky: Teichmuller geodesics in {K}

hence hyperbolic arguments apply.

For the remainder, there is an analogue of vertices of infinite degree, subsurfaces of {\Sigma}. 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 {G}) plays the role of the hyperbolic graph {Y}.

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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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