## Notes of Erik Guentner’s Cambridge lecture 23-05-2017

Affine actions, cohomology and hyperbolicity

When can a group act properly on a Hilbert space or an ${L^p}$ space? I start from scratch.

1. Affine actions

${G}$ a discrete group, ${E}$ a Banach space. We are interested in actions of ${G}$ on ${E}$ where each element ${\alpha(g)}$ is an affine transformation whose linear part is isometric. Equivalently, one is given an linear representation ${\pi:G\rightarrow O(E)}$ and a cocycle ${b:G\rightarrow E}$, and

$\displaystyle \begin{array}{rcl} \alpha(g)=\pi(g)+b(g). \end{array}$

Such cocycles ${b\in Z^1(G,E)}$ represent classes in ${H^1(G,E)}$. Coboundaries give equivalent actions (merely change origin).

My point: typically, ${E=\ell_p(S)}$, ${S}$ a ${G}$-set, and coboundaries ${\phi-\pi_g(\phi)\in \ell_p(S)}$, for some ${\phi\in\ell_\infty(S)}$ which need not belong to ${\ell_p(S)}$.

The action is proper if ${|b(g)|\rightarrow\infty}$ as ${|g|\rightarrow\infty}$.

2. Hyperbolicity

2.1. Example: trees

${X=}$ simplicial tree. Let ${G}$ act on ${X}$. For vertex ${v}$, define ${\mu_v(x)\in\ell_1(lk(x))}$ as follows. If ${a}$ is a neighbour of ${x}$, set ${\mu_v(x)(a)=1}$ wether ${x}$ belongs to the arc ${[v,a]}$ or not.

This defines an action of ${G}$ on ${\ell_1(TX)}$, where ${TX}$ (the “tangent bundle of the tree”) is the union of all links of vertices, where the cocycle is

$\displaystyle \begin{array}{rcl} b(g)=\phi-g\circ\phi \end{array}$

where ${\phi=\mu_v}$, for a fixed ${v}$. Then ${\phi\in\ell_\infty(TX)}$. Nevertheless,

$\displaystyle \begin{array}{rcl} \|b\|_{\ell_1}=2 d(v,gv). \end{array}$

Hence it is proper.

2.2. Example: negatively curved manifolds

See the book by Nowak and Yu. ${\mu_v}$ become a unit vector field, the gradient of distance to ${v}$. The norm of ${\mu_v(x)-\mu_{gv}(x)}$ decays exponentially with ${d(x,v)}$ for fixed ${g}$ (in the tree case, it fell instantly from 1 to 0). If curvature is bounded below, balls grow exponentially, hence ${b(g)\in L^p(TM)}$. Every point in a tube around the geodesic segment between ${v}$ and ${gv}$ contributes to the norm, hence the ${L^p}$-norm is proportional to ${d(v,gv)}$, showing properness.

2.3. Example: hyperbolic groups

Inspired from Yu’s treatment. Let us define ${\mu_v(x)\in\ell_1(B(x,4\delta))\cap[x,v]_{2\delta}}$, where

$\displaystyle \begin{array}{rcl} [x,v]_{2\delta}=\{y\,;\,d(x,y)+d(y,v)\leq d(x,v)+2\delta\}. \end{array}$

What I describe corresponds to ${-\mu_v(x)}$ in the previous example: pointing towards ${v}$ instead of opposite to ${v}$. Move along a geodesic from ${x}$ to ${v}$. Iteratively average

Again, the key points are

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

$\displaystyle \begin{array}{rcl} \|\mu_v(x)-\mu_w(x)\|_{\ell_1}\leq C\,e^{-\epsilon d(x,v)}. \end{array}$

3. Relative hyperbolicity

${G}$, ${P finitely generated groups. Define the combinatorial horoball over ${P}$ as follows. Start with infinitely many copies of ${Cay(P,S)}$ called layers, sitting in ${Cay(P\times{\mathbb Z},S\cup{1}). On }$n${-th layer, add edges between points at distance }$n$latex {. Attach such combinatorial horoballs over the cosets of }&fg=000000$P${ to }$Cay(G,S_G)${. The resulting graph }$X${ is }$\delta$latex {-hyperbolic. Unfortunately, is does not have bounded geometry, this kills Mineyev estimates. \nbegin{theorem}{Guentner-Reckwerdt-Tessera} Assume }&fg=000000$P${ has polynomial growth. Then }$G${ admits a proper affine action on a mixed }$\ell_p-\ell_1${-space, and also on an }$L^p$latex { space. \end{theorem} In Yu’s treatment, passing from }&fg=000000$\ell_p-\ell_1${ to }$L^p$latex { follows from bounded geometry. \subsection{Proof} The new trick has independent interest. \begin{enumerate} \item Exponential growth of balls with center in the base. \item Existence of thinned horoballs. \item Averaging technique. \end{enumerate} By a thinned horoball, I mean a bounded geometry subset }&fg=000000$\theta${ of the combinatorial horoball, obtained by decimating vertices in slices. It is still quasi-isometric to the combinatorial horoball. Unfortunately, the }$P${-action is lost. It is thinned with respect to a choice of parameters. Let }$\Theta${ denote the set of thinned horoballs with fixed parameters. It is compact in pointed Gromov-Hausdorff topology. Hence there is a }$P${-invariant probability measure. A thinning of the whose cusped space is a choice of thinned horoball in each glued combinatorial horoball }$(\theta_t)_{t\in G/P}${, }$\theta_t\in \Theta${. The infinite product of }$P${-invariant measure is a }$G\$-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