Combination theorems in convex projective geometry

TL;DR

This paper develops a general combination theorem for convex projective groups, enabling new constructions and properties of discrete subgroups in projective geometry, with applications to linearity, Zariski density, and convex cocompactness.

Contribution

It introduces a novel combination theorem for convex projective groups, extending classical results and applying to linearity, Zariski density, and convex cocompactness in higher-rank geometries.

Findings

Free product of two linear groups remains linear.

Constructs Zariski-dense discrete subgroups not lattices.

Free product of convex cocompact groups is convex cocompact.

Abstract

We prove a general combination theorem for discrete subgroups of $\mathrm{PGL}(n,\mathbb{R})$ preserving properly convex open subsets in the projective space $\mathbb{P}(\mathbb{R}^n)$, in the spirit of Klein and Maskit. We use it in particular to prove that a free product of two $(\mathbb{Z}$-)linear groups is again ($\mathbb{Z}$-)linear, and to construct Zariski-dense discrete subgroups of $\mathrm{PGL}(n,\mathbb{R})$ which are not lattices but contain a lattice of a smaller higher-rank simple Lie group. We also establish a version of our combination theorem for discrete groups that are convex cocompact in $\mathbb{P}(\mathbb{R}^n)$ in the sense of arXiv:1704.08711. In particular, we prove that a free product of two convex cocompact groups is convex cocompact, which implies that the free product of two Anosov groups is Anosov. We also prove a virtual amalgamation theorem over convex cocompact subgroups generalizing work of Baker-Cooper.