Filling functions and non-positive curvature
Continuation of Leuzinger’s talk. I will bring support to his conjecture and explain why it is so hard. And then describe some preliminary results.
is the semi-direct product of by acting by diagonal matrices of determinant 1.
has exponential Dehn function whereas has quadratic Dehn function.
is a horosphere in . In , small disks filling a given closed curve are essentially unique. If I insist that the disk avoids a ball of radius , this will have an exponentially large cost. Idem if one must avoid a horoball (note that the intersection of flat with a horoball is a diamond shaped compact set).
is a horosphere in . The intersection of a flat with a horoball is a octahedron. Now, a curve in the complement can be filled with a disk going around the octahedron, this may yield a quadratic 1-filling.
In general, we expect -filling in to be easy for but hard for .
2. Non-uniform lattices
Drutu: -rank 1 lattices have .
Young: has quadratic Dehn function if .
Bestvina-Eskin-Wortman: -arithmetic lattices with have polynomial Dehn functions.
Theorem 1 In , for ,
2.1. I explain it for
First show that a curve is covered by several flats.
Lemma 2 (Simple edges) Any two points are connected by a curve contained in a single flat. (avoid the octahedron).
Lemma 3 (Simple triangles) A curve in a finite union of flats can be filled with a disk in a larger finite union of flats.
Then, an arbitrary curve can be broken into triangles. Adding up areas yields . That’s all for .
2.2. In higher dimensions
It is harder to break down a sphere into simple spheres. Diameter could be arbitrarily large. In fact, Lipschitz spheres split nicely but to handle general cycles, we use Whitney decompositions (Lang and Schlichtenmaier).
We have a very general method which turns Lipschitz extension theorems into filling inequalities. The general setting is a metric space with finite Assouad-Nagata dimension, and a subset in . We assume a filling inequality in and a Lipschitz extension property for (Lipschitz spheres extend to the disk with a controlled inscrease in Lipschitz constant). Finite Assouad-Nagata dimension allows to construct a covering of by balls with radii comparable to distance to and bounded multiplicity (Whitney decomposition). Let be the nerve of this covering. Given a cycle in , it spans a chain in . Approximate by a cycle in a skeleton of (à la Federer-Fleming). To project to , first project arbitrarily vertices of to nearest points on . Since edges in have length proportional to their distance to , this projection is again Lipschitz. Apply Lipschitz extension property to get extension to 1-skeleton, and so on.
The Whitney decomposition trick was already used in the proof of the quadratic Dehn function for .