** Affine actions, cohomology and hyperbolicity **

When can a group act properly on a Hilbert space or an space? I start from scratch.

**1. Affine actions **

a discrete group, a Banach space. We are interested in actions of on where each element is an affine transformation whose linear part is isometric. Equivalently, one is given an linear representation and a cocycle , and

Such cocycles represent classes in . Coboundaries give equivalent actions (merely change origin).

My point: typically, , a -set, and coboundaries , for some which need not belong to .

The action is *proper* if as .

**2. Hyperbolicity **

** 2.1. Example: trees **

simplicial tree. Let act on . For vertex , define as follows. If is a neighbour of , set wether belongs to the arc or not.

This defines an action of on , where (the “tangent bundle of the tree”) is the union of all links of vertices, where the cocycle is

where , for a fixed . Then . Nevertheless,

Hence it is proper.

** 2.2. Example: negatively curved manifolds **

See the book by Nowak and Yu. become a unit vector field, the gradient of distance to . The norm of decays exponentially with for fixed (in the tree case, it fell instantly from 1 to 0). If curvature is bounded below, balls grow exponentially, hence . Every point in a tube around the geodesic segment between and contributes to the norm, hence the -norm is proportional to , showing properness.

** 2.3. Example: hyperbolic groups **

Inspired from Yu’s treatment. Let us define , where

What I describe corresponds to in the previous example: pointing towards instead of opposite to . Move along a geodesic from to . Iteratively average

Again, the key points are

- Bounded geometry.
- Exponential decay of information (Mineyev):

**3. Relative hyperbolicity **

, finitely generated groups. Define the combinatorial horoball over as follows. Start with infinitely many copies of called layers, sitting in nn$latex {.

Attach such combinatorial horoballs over the cosets of }&fg=000000$PCay(G,S_G)X\delta$latex {-hyperbolic. Unfortunately, is does not have bounded geometry, this kills Mineyev estimates.

\nbegin{theorem}{Guentner-Reckwerdt-Tessera} Assume }&fg=000000$PG\ell_p-\ell_1L^p$latex { space. \end{theorem}

In Yu’s treatment, passing from }&fg=000000$\ell_p-\ell_1L^p$latex { follows from bounded geometry.

\subsection{Proof}

The new trick has independent interest.

\begin{enumerate} \item Exponential growth of balls with center in the base. \item Existence of thinned horoballs. \item Averaging technique. \end{enumerate}

By a thinned horoball, I mean a bounded geometry subset }&fg=000000$\thetaP\ThetaP(\theta_t)_{t\in G/P}\theta_t\in \ThetaPG$-invariant probability measure in the space of thinnings. Apply the method of Mineyev-Yu, Lafforgue to each thinning and then average.

**4. Questions **

Schwartz: you could weight the edges instead? We tried but could not make it work.

Breuillard: what is the growth