** Spheres in horospheres **

**1. The result **

**Conjecture**. -filling is exponential for non uniform irreducible lattices in rank lattices.

Thurston did it for . Wortman solved the conjecture in most cases for simple ambient groups, when the -rank is maximal.

**Theorem 1 (Leuzinger-Pittet)** * Conjecture is ok for types , , , . *

Given linear forms on , say is separated from others if on ker, is separated from others. And in dimension 1, is separated from if they have opposite signs.

**Proposition 2** * Let
*

*
* contain the extension of by , semi-direct product defined by linear forms . Assume that

1. Inclusion is at most polynomially distorted.

2. is separated from others.

*
Then for any , and all , there exists an -sphere in of volume whose filling volume in is . *

**2. Proof **

If , this is Gromov’s construction in . If , view as a -bundle over . Draw a one-parameter family of Gromov curves.

Let be a semi-simple -algebraic group, a maximal -split torus, pick a basis of simple roots.

**Proposition 3** * Assume -rank-rank and that is of type , or . One can choose simple roots , positive roots and pairwise commuting vectors in such that *

*
*
1. generates a unipotent subgroup.

*
2. The separation property is satisfied. *

**Example 1** * Let , . The roots are edge-mid-points of a cube. , and the origin form an equilateral triangle in a plane . , and are edge mid-points on different faces of the cube. Their restrictions to P are equilaterally situated. *

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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/