Spaces admitting thin triangles and median spaces
Joint work with Cornelia Drutu and Frederic Haglund, initiated 10 years ago.
Theorem 1 Let 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 2 Lattices 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 3 A 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 4 Given a measured wall space, defined the wall (pseudo-)metric
Example. In or , one recovers the Euclidean (resp. hyperbolic) metric.
Proposition 5 Any 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 6 Let be a convex set. Let , let be the projection of to (-projection suffices). Then