Notes of Christophe Pittet’s lecture

Spheres in horospheres

1. The result

Conjecture. {r}-filling is exponential for non uniform irreducible lattices in rank {r} lattices.

Thurston did it for {Sl(r+1,{\mathbb Z})}. Wortman solved the conjecture in most cases for simple ambient groups, when the {\mathbb{Q}}-rank is maximal.

Theorem 1 (Leuzinger-Pittet) Conjecture is ok for types {A_r}, {B_r}, {C_r}, {BC_r}.

Given linear forms {\lambda_1,...,\lambda_{n+1}} on {\mathbb{R}^n}, say {\lambda_{n+1}} is separated from others if on ker{(\lambda_{n+1})}, {\lambda_n} is separated from others. And in dimension 1, {\lambda_2} is separated from {\lambda_1} if they have opposite signs.

Proposition 2 Let

\displaystyle  \begin{array}{rcl}  1 \rightarrow N \rightarrow G \rightarrow \mathbb{R}^{r-1} \rightarrow 1 \end{array}

contain the extension {Sol_A} of {\mathbb{R}^r} by {\mathbb{R}^{r-1}}, semi-direct product defined by linear forms {\lambda_1,..,\lambda_r}. Assume that

1. Inclusion {\mathbb{R}^r \rightarrow N} is at most polynomially distorted.

2. {\lambda_r} is separated from others.

Then for any {\delta>0}, and all {v>1}, there exists an {r-1}-sphere in {Sol_A} of volume {< v} whose filling volume in {G} is {> \exp(\delta v^{1/r-1})}.

2. Proof

If {r=2}, this is Gromov’s construction in {Sol_3}. If {r=3}, view {Sol_5} as a {Sol_3}-bundle over {\mathbb{R}^2}. Draw a one-parameter family of Gromov curves.

Let {G} be a semi-simple {{\mathbb Q}}-algebraic group, {D} a maximal {{\mathbb Q}}-split torus, pick a basis of simple roots.

Proposition 3 Assume {\mathbb{Q}}-rank{(G)= \mathbb{R}}-rank{(G)} and that {G} is of type {A_r}, {C_r} or {BC_r}. One can choose simple roots {a_1,...,a_{r-1}}, positive roots {\lambda_1,...,\lambda_r} and pairwise commuting vectors {v_i} in {\mathfrak{g}_{\lambda_i}} such that

1. {\mathbb{R}^r = <v_1,...,v_r>} generates a unipotent subgroup.

2. The separation property is satisfied.

Example 1 Let {G=Sl(4)}, {r=3}. The roots are edge-mid-points of a cube. {a_1}, {a_2} and the origin form an equilateral triangle in a plane {P}. {\lambda_1}, {\lambda_2} and {\lambda_3} are edge mid-points on different faces of the cube. Their restrictions to P are equilaterally situated.

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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