** Spaces admitting thin triangles and median spaces **

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

Theorem 1Let be a space with walls. Assume that is -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 .

Corollary 2Lattices in groups 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 cube complex. It turns out this is not the case: with Fernos and Iozzi, we proved that cocompact lattices in 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

Say that is *median* if for every triangle, the three intervals intersect at a single point . is called the median.

**Examples**. , -trees, , , -cube complexes with -metric on cubes.

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

**Examples**. Hyperplanes in -cube complexes with counting measure. All hyperplanes in with the natural motion-invariant measure. All hyperplanes in hyperbolic space with the natural motion-invariant measure.

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

**Example**. In or , one recovers the Euclidean (resp. hyperbolic) metric.

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

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

**Example**. Fix . The section which associates to a wall the half-space containing is admissible.

Let be the set of admissible sections. Let be the subset of sections such that has finite measure. Then inherit a metric, it is median.

**2. Spaces admitting thin triangles **

Say a metric space admits thin triangles if such that for any 3 points , there exist on intervals such that diameter.

**Example**. Hyperbolic metric spaces and groups, median spaces, products of those. However, Euclidean spaces do not admit thin triangles.

**Example**. A tripod of 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 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 admits thin triangles. We prove that is at finite Hausdorff distance of the image of .

Here is the key step.

Lemma 6Let be a convex set. Let , let be the projection of to (-projection suffices). Then