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 .
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.