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.

2. Spaces admitting thin triangles

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}

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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