## Notes of Romain Tessera’s Lille lecture nr 1

Hyperbolic Lie groups

1. Non vanishing of ${\ell^p}$ cohomology

Theorem 1 Let ${X}$ be a connected graph with bounded degree, which is hyperbolic. Assume that ${X}$ contains en quasi-isometrically embedded 3-regular tree ${T}$. Then ${\ell^p\bar{H}^1(X)\not=0}$ for ${p}$ lage enough.

To simplify the proof, I will use an extra assumption: Given a base point ${o\in X}$, ${\exists C}$ such that every point sits at distance at most ${C}$ from a geodesic ray passing through ${o}$.

${\partial T}$ is a Cantor set embedded in ${\partial X}$. Let ${O_1}$ and ${O_2}$ be disjoint open sets that both intersect ${\partial T}$ an together cover ${\partial T}$. Consider the function on ${\partial X}$

$\displaystyle \begin{array}{rcl} F(x)=\max\{0,1-\frac{1}{t}d(x,O_1), \end{array}$

where ${t=d(O_1,O_2)>0}$. Extend ${F}$ radially to a function ${f}$ on ${X}$. Then ${f\in D^p}$ for ${p}$ large enough. Indeed, ${df(e)}$ decays exponentially in terms of the distance of edge ${e}$ to ${o}$. Its restriction belongs to ${D^p(T)}$. It pairs non trivially with the ${\ell^{p'}}$ cycle supported on ${T}$ which takes value 1 on an edge, ${1/2}$ on neighboring edges, ${1/4}$ on the next layer, and so on. Therefore its reduced cohomology class does not vanish.

2. ${L^p}$ cohomology of locally compact groups

Let ${G}$ be locally compact, compactly generated. Fix a compact generating set. This turns ${G}$ into a metric space.

Fact. There exists a connected bounded degree graph ${X}$ which is quasiisometric to ${G}$.

Indeed, pick a maximal set of disjoint unit balls in ${G}$, these are the vertices, and put edges when centers are at distance ${\leq 4}$.

Let

$\displaystyle \begin{array}{rcl} D^p(G)=\{f:G\rightarrow{\mathbb R}\,;\,\forall g\in G,\,f-\rho(g)f\in L^p(G)\}. \end{array}$

Define

$\displaystyle \begin{array}{rcl} \bar{H}^1_p(G)=D^p(G)/\overline{L^p(G)+{\mathbb R}}. \end{array}$

Gromov showed that non elementary hyperbolic groups contain quasiisometrically embedded trees. The argument (ping-pong) extends to compactly generated groups.

Corollary 2 If ${G}$ is hyperbolic as a metric space, then ${\bar{H}^1_p(G)\not=0}$ for ${p}$ large enough.

3. Hyperbolic Lie groups

Question. Which connected Lie groups are hyperbolic ?

Example 1 Heintze groups, the affine group of the line (it is Heintze), the affine group of ${{\mathbb Q}_p}$, ${Sl(2,{\mathbb R})}$, ${Sl(2,{\mathbb Q}_p)}$.

Note these groups either act transitively on negatively curved manifolds, or cocompactly on trees. We shall prove that this is general.

3.1. Reduction

Let ${G}$ be a connected Lie group (over ${{\mathbb R}}$ or ${{\mathbb Q}_p}$). There exists a cocompact closed subgroup ${H and a compact normal subgroup ${W< H}$ such that ${B=H/W}$ satisfies

$\displaystyle \begin{array}{rcl} 1\rightarrow U\rightarrow B\rightarrow {\mathbb Z}^d \rightarrow 1 \end{array}$

where ${U}$ is simply connected nilpotent.

Note that one really needs ${{\mathbb Z}^d}$ and not ${{\mathbb R}^d}$.

3.2. Classification

Theorem 3 (Cornulier-Tessera) Let ${G}$ be a connected Lie group (over ${{\mathbb R}}$ or ${{\mathbb Q}_p}$). The following are equivalent.

1. ${G}$ is hyperbolic.
2. ${B}$ is Heintze in the sense that ${B=U\times {\mathbb Z}}$ and the action of ${{\mathbb Z}}$ on ${U}$ is contracting.
3. ${\bar{H}^1_p(G)\not=0}$ for some ${p>1}$.
4. ${\bar{H}^1_p(G)\not=0}$ for ${p}$ large enough.

We have only partial results for general compactly generated groups: we can treat (with Caprace and Monod) the case when ${G}$ has an amenable cocompact subgroup.

Note that such a classification fails for discrete groups. For instance, ${{\mathbb Z}^2\star{\mathbb Z}}$ has nonzero reduced ${\ell^p}$ cohomology for all ${p}$ (it has infinitely many ends) but it is not hyperbolic.