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<G} 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.


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


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
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