Notes of Indira Chatterji’s Rennes lecture

${CAT(0)}$-cube complexes and the median class

Joint with Talia Fernos and Alessandra Iozzi.

1. Motivation

The following corollary.

Theorem 1 A cocompact, irreducible lattice in ${Sl(2,{\mathbb R})\times Sl(2,{\mathbb R})}$ is not cubical.

Completed by Fernos, Caprace, Lecureux in order to prove that anay such lattives, when acting isometrically on a ${CAT(0)}$ cube complex, must have a fixed point.

Consider lattices in semi-simple Lie groups ${G}$.

If ${G}$ has property (T), they can’t be cubical. If ${G=SO(3,1)}$, they are cubical (Bergeron-Wise, using Kahn-Markovic, unobvious). For ${SO(4,1)}$, some lattices are cubical, as Anne Giralt explained, but for the other ones we don’t know.

2. Main result

Theorem 2 Let ${\Gamma}$ act isometrically on a ${CAT(0)}$ cube complex ${X}$ in a non elementary manner (no fixed point on ${X}$ nor on the visual boundary ${\partial X}$). Then a certain bounded cohomology class ${m\in H_b^2(\Gamma,\pi)}$ vanishes.

Corollary 3 (Superrigidity) Let ${\Gamma}$ be a cocompact irreducible lattice in a product of locally compact groups ${G}$. Let ${\Gamma}$ act essentially and non-elementarily on a ${CAT(0)}$ cube complex. Then the action extends continuously to ${G}$, factoring via one of the factors.

The fact that this follows from the theorem is due to Shalom and Burger-Monod.

3. The median class

3.1. Case of trees

I explain the case when ${X}$ is a tree.

Let ${H}$ be the set of oriented paths of length 2 in the tree. Let ${\pi}$ be the obvious action of ${\Gamma}$ on ${\ell^2(H)}$. Let ${w:X\times X\rightarrow \ell^2(H)}$ be defined by

$\displaystyle \begin{array}{rcl} w(x,y)=1_{[[x,y]]}-1_{[[y,x]]}, \end{array}$

where ${[[x,y]]}$ denotes the set of ${a\in H}$ which are between ${x}$ and ${y}$. This is unbounded, but the coboundary ${dw}$ is bounded. Indeed, cancellations leave us only with paths that touch the median.

In the case of trees, this is not surprising (classical fact that generalizes to hyperbolic metric spaces).

3.2. Median metric spaces

In a metric, the side ${I(x,y)}$ is the set of points for wich the triangle inequality is an equality. A metric space is median if given 3 points, there is a unique common point to the 3 sides.

${CAT(0)}$ cube complexes equipped with the metric which is ${\ell^1}$ on cubes are median.

3.3. Case of CCC

${CAT(0)}$ cube complexes have half-spaces: start cutting s cube in equal parts and continue for ever in contiguous cubes. Say half-spaces ${h_1 \subset h_2}$ are tightly nested if any half-space that sits in between must be one of them. Define ${H}$ as the set of pairs of tightly nested half-spaces. The same formula defines a 1-cochain ${w}$. The same cancellations show that ${dw}$ only involves pairs touching the median point. Therefore it is bounded. However, the bound depends on the dimension of ${X}$. Pull-back ${dw}$ on ${\Gamma}$ via an orbit.

3.4. Non vanishing

Burger-Monod show that

$\displaystyle \begin{array}{rcl} H_b^2(\Gamma,\pi)\equiv ZL_{alt,*}^{\infty}(B,\pi)^{\Gamma}, \end{array}$

where ${B}$ is a Poisson boundary. We use the Roller compactification, defined as follows. ${X}$ embeds in the set of subsets of ${H}$ (a point is mapped to the set of half-space pairs that contain it). Take the closure of the image of that embedding. Then (Zimmer), there is an equivariant map of ${B}$ to the set of probability measures on ${\bar{X}}$. One shows that the image is in ${\partial X}$, this gives the image of ${dw}$ as a nonzero cocycle on ${B}$.