## 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 ${Y}$.