## 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. ${\mathbb{R}^r = }$ 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.