Exemples de vari\'et\'es projectives strictement convexes de volume fini en dimension quelconque

TL;DR

This paper constructs examples of finite-volume, non-compact, strictly convex projective manifolds in all dimensions, along with Zariski-dense discrete subgroups of SL(n+1,R) that are neither lattices nor Schottky groups, and shows the convex sets are Gromov-hyperbolic.

Contribution

It provides new examples of finite-volume, non-hyperbolic, strictly convex projective manifolds in arbitrary dimensions and constructs Zariski-dense discrete subgroups with novel properties.

Findings

Constructed non-compact, finite-volume convex projective manifolds in all dimensions.

Built Zariski-dense discrete subgroups of SL(n+1,R) that are not lattices or Schottky groups.

Showed the convex sets are Gromov-hyperbolic.

Abstract

We build examples of properly convex projective manifold $\Omega/ \Gamma$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$ which are not lattice, nor Schottky groups. Moreover, the open properly convex set $\Omega$ is strictly-convex, even Gromov-hyperbolic. Nous construisons des exemples de vari\'et\'es projectives $\Omega/ \Gamma$ proprement convexes de volume fini, non hyperbolique, non compacte en toute dimension $n \geqslant 2$. Ceci nous permet au passage de construire des groupes discrets Zariski-dense de $\SL_{n+1}(\R)$ qui ne sont ni des r\'eseaux de $\SL_{n+1}(\R)$, ni des groupes de Schottky. De plus, l'ouvert proprement convexe $\Omega$ est strictement convexe, m\^eme Gromov-hyperbolique.