## Notes of Romain Tessera’s Southampton lecture 27-03-2017

Proper ${L^p}$-actions and weak amenability for relatively hyperbolic groups

1. Weak amenability

Start with the following characterization of amenability. ${G}$ is amenable iff there is a sequence of finitely supported positive definite functions ${\phi_n}$ tending to 1 pointwise as ${n}$ tends to ${\infty}$.

We can relax finitely supported into ${c_0}$. This gives Haaerup’s property.

We can instead relax positive definite into ${\|\phi_n\|_{CB}}$ bounded. This gives weak amenability.

Positive definite functions coincide with coefficients of unitary representations (for instance, characteristic functions of Folner sets in the regular representation).

Completely bounded functions arise as ${\phi(gh^{-1})=\langle\xi(g),\eta(h)\rangle}$, where ${\xi}$ and ${\eta}$ are maps to some Hilbert space with ${\|\xi(g)\|\leq\sqrt{\Lambda}}$, ${\|\eta(g)\|\leq\sqrt{\Lambda}}$.

Therefore Amenability implies Haagerup (and thus existence of a proper affine isometric action on a Hilbert space). Also, amenability implies weak amenability with constant ${\Lambda=1}$ (and thus weak amenability with an arbitrary constant ${\Lambda}$).

1.1. General results

Yu: Hyperbolic groups act properly on ${L^p}$ spaces for ${p}$ large enough.

Ozawa: Hyperbolic groups are weakly amenable.

1.2. A class of groups for which much is known

Theorem 1 (Haagerup) If ${\Gamma}$ is a lattice in a simple Lie group, then ${\Gamma}$ is weakly amenable iff rank is 1. Constant ${\Lambda}$ is 1 for ${SO(n,1)}$ and ${SU(n,1)}$, ${2n-1}$ for ${Sp(n,1)}$ and

As far as proper ${L^p}$-actions are concerned, higer rank lattices have fixed points on ${L^p}$-spaces. ${SO(n,1)}$ and ${SU(n,1)}$ are Haagerup. ${Sp(n,1)}$ acts properly on ${L^p}$ iff ${p>4n+2}$.

Free groups are Haagerup and weakly amenable with ${\Lambda=1}$.

2. Permanence properties

2.1. Free product

Haagerup, proper ${L^p}$-actions and weak amenability with ${\Lambda=1}$ (Jolissaint, Ricard-Xu).

2.2. Graph product

Haagerup, proper ${L^p}$-actions and weak amenability with ${\Lambda=1}$ (Antolin-Dreesen, Rechwerdt).

Theorem 2 (Guentner-Rechwerdt-Tessera) If ${G}$ is relatively hyperbolic with parabolic subgroups of polynomial growth, then ${G}$ is weakly amenable and acts properly on ${L^p}$.

The following example does not follow from earlier results.

Corollary 3 If ${G}$ is hyperbolic, ${G\star {\mathbb Z}^2}$ is weakly amenable.

(In fact, it does, since weak amenability is preserved under measure-equivalence, and Gaboriau’s theorem implies that free product with an amenable group preserves measure equivalence, dixit Ozawa).

3. Proof of Yu’s theorem, according to Alvarez-Lafforgue

As a baby case, assume ${G=F_2}$ is free. Let ${X}$ denote the usual Cayler graph. We design a map

$\displaystyle \begin{array}{rcl} X\times X\rightarrow \mathcal{P}(X),\quad (x,a)\mapsto \mu_x(a), \end{array}$

as follows: ${\mu_x(a)}$ is the Dirac measure on the neighbour of ${a}$ closest to ${x}$. then

1. The support of ${\mu_x(a)}$ is contained in ${B(a,1)}$,
2. ${\|\mu_x(a)-\mu_{x'}(a)\|=2}$ for ${a\in[x,x']}$, ${=0}$ otherwise,
3. ${\mu_x(a)}$ and ${\mu_{x'}(a)}$ have disjoint support iff ${a\in[x,x']}$.

Form a cocycle ${c(g)=\mu_1-\mu_g\in\ell^p(X^{\leq 1})}$. Then ${\|c\|_p\sim |g|^{1/p}}$.

Theorem 4 (Mineyev, Alvarez-Lafforgue) Let ${X}$ be a ${\delta}$-hyperbolic bounded degree graph. There exists ${C,\epsilon>0}$, there exists a family of measures such that

1. The support of ${\mu_x(a)}$ is contained in ${B(a,4\delta)}$,
2. ${\|\mu_x(a)-\mu_{x'}(a)\|\leq C e^{-\epsilon d(x,a)}}$ for ${d(x,x')\leq 1}$,
3. ${\mu_x(a)}$ and ${\mu_{x'}(a)}$ have disjoint support for a number of values of ${a}$ which is at least ${\epsilon d(x,x')-C}$..

Form a cocycle ${c(g)=\mu_o-\mu_g.o\in\ell^p(X^{\leq 4\delta})}$ for ${p}$ large enough (depending on volume growth).

4. The relative hyperbolic case.

The above argument required bounded degree. Beware that the coned-off graph of relatively hyperboliv does not have this property.

The Groves-Manning space (close to an idea of Gromov’s) is the Cayley graph together with combinatorial horoballs glued to it along the cosets of parabolic subgroups. Combinatorial horoball is coset of parabolic subgroup ${P}$ times ${{\mathbb N}}$ with edges joining vertices of distance ${n}$ on the same level ${n}$. The resulting space ${Y}$ is ${\delta}$-hyperbolic.

If ${P}$ has polynomial growth, ${Y}$ has exponential growth, and one finds ${p}$ large enough that the cocycle is bounded in ${L^p}$.

${Y}$ does not have bounded degree. Fortunately, if ${P}$ has polynomial growth, is is quasi-isoperimetric to a graph of bounded geometry. But we loose equivariance. The space of graphs of degree ${\leq k}$, ${(C,K)}$quasi-isometric to ${Y}$ is compact. Does it carry a ${G}$-invariant measures?

Pick a ${P}$-invariant measure ${\nu}$ and let ${\mu=\nu^{\otimes G/P}}$. This is the required invariant measure.