## Notes of Indira Chatterji’s Cambridge lecture 13-04-2017

Spaces admitting thin triangles and median spaces

Joint work with Cornelia Drutu and Frederic Haglund, initiated 10 years ago.

Theorem 1 Let ${X}$ be a space with walls. Assume that ${X}$ is ${\mu}$-locally finite. Assume that the wall metric has thin triangles. Then the median space associated to the wall structure is at finite distance Hausdorff distance from ${X}$.

Corollary 2 Lattices in groups ${SO(n,1)}$ and in products of copies of such act isometrically and properly, with bounded quotient, on a median space.

10 years ago, we hoped to deduce an action on a ${CAT(0)}$ cube complex. It turns out this is not the case: with Fernos and Iozzi, we proved that cocompact lattices in ${SO(n,1)\times SO(n,1)}$ are not cubulable (by bounded cohomology superrigidity). So the corollary is the best one can expect.

1. Media spaces and spaces with walls

In a geodesic metric space, the interval between two points is

$\displaystyle \begin{array}{rcl} I(x,y)=\{z\in X\,;\,d(x,z)+d(z,y)=d(x,y)\}. \end{array}$

Say that ${X}$ is median if for every triangle, the three intervals intersect at a single point ${m}$. ${m}$ is called the median.

Examples. ${{\mathbb R}}$, ${{\mathbb R}}$-trees, ${\ell^1}$, ${L^1}$, ${CAT(0)}$-cube complexes with ${\ell^1}$-metric on cubes.

Definition 3 A measured wall space is a set ${X}$ equipped with a collection ${\mathcal{H}}$ of subsets (called half-spaces) stable under complementation, with a measure ${\mu}$ on ${\mathcal{H}}$, such that the measure of the subset of half-spaces separating two points is always finite.

Examples. Hyperplanes in ${CAT(0)}$-cube complexes with counting measure. All hyperplanes in ${{\mathbb R}^n}$ with the natural motion-invariant measure. All hyperplanes in hyperbolic space ${H^n}$ with the natural motion-invariant measure.

Definition 4 Given a measured wall space, defined the wall (pseudo-)metric

$\displaystyle \begin{array}{rcl} d(x,y)=\mu(w(x|y)). \end{array}$

Example. In ${{\mathbb R}^n}$ or ${H^n}$, one recovers the Euclidean (resp. hyperbolic) metric.

Proposition 5 Any wall metric embeds isometrically in a median space.

Indeed, consider sections, i.e. maps ${s}$ assigning to each wall one of the 2 a half-spaces it bounds. Say ${s}$ is admissible if disjoint walls are never mapped to opposite half-spaces.

Example. Fix ${x_0\in X}$. The section ${s_{x_0}}$ which associates to a wall the half-space containing ${x_0}$ is admissible.

Let ${\overline{M(X)}}$ be the set of admissible sections. Let ${M(X)}$ be the subset of sections ${s}$ such that ${s\Delta s_{x_0}}$ has finite measure. Then ${M(X)}$ inherit a metric, it is median.

Say a metric space admits thin triangles if ${\exists \delta}$ such that for any 3 points ${x,y,z}$, there exist ${x',y',z'}$ on intervals such that diameter${(x',y',z')\leq\delta}$.

Example. Hyperbolic metric spaces and groups, median spaces, ${\ell^1}$ products of those. However, Euclidean spaces do not admit thin triangles.

Example. A tripod of ${\ell^1}$ planes has thin triangles but it is not median.

Not much is known about such spaces. They are expected to have quadratic filling, but only a sub-cubic bound is known.

3. Sketch of proof

Assume ${X}$ is a space with walls. Assume that the measure of the set of walls intersecting a ball is bounded above, uniformly in terms of its radius. Assume that ${X}$ admits thin triangles. We prove that ${M(X)}$ is at finite Hausdorff distance of the image of ${X}$.

Here is the key step.

Lemma 6 Let ${C\subset X}$ be a convex set. Let ${x\in X}$, let ${p}$ be the projection of ${x}$ to ${C}$ (${\epsilon}$-projection suffices). Then

$\displaystyle \begin{array}{rcl} \mu(w(x|p)\setminus w(x|C))\leq 2\delta+\epsilon. \end{array}$