Abstract
In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.
Highlights
The Lie group PGLd+1(R) acts naturally on real projective space P(Rd+1) and for an open set Ω ⊂ P(Rd+1) we define the automorphism group of Ω asAut(Ω) = {φ ∈ PGLd+1(R) : φΩ = Ω}.An open set Ω is called a convex divisible domain if it is a bounded convex open set in some affine chart of P(Rd+1) and there exists a discrete group Γ ≤ Aut(Ω) which acts properly, freely, and cocompactly on Ω.The fundamental example of a convex divisible domain comes from the KleinBeltrami model of real hyperbolic d-space HdR
This convex divisible domain is a model of the symmetric space associated to the simple Lie group SO(p, p)
In contrast to the many examples of convex divisible domains in real projective space, we prove that every convex divisible domain in Grp(R2p) is symmetric and even more precisely that up to projective isomorphism Bp,p is the only convex divisible domain in Grp(R2p)
Summary
The Lie group PGLd+1(R) acts naturally on real projective space P(Rd+1) and for an open set Ω ⊂ P(Rd+1) we define the automorphism group of Ω as. Suppose p > 1, Ω ⊂ Grp(R2p) is a bounded convex open subset of some affine chart, and there exists a discrete group Γ ≤ Aut(Ω) so that Γ acts cocompactly on Ω. In the final part of the proof we show that Aut0(Ω), the connected component of the identity of Aut(Ω), is a simple Lie group which acts transitively on Ω Our approach for this step is based on work of Farb and Weinberger [FW08] who prove a number of remarkable rigidity results for compact aspherical Riemannian manifolds whose universal covers have non-discrete isometry groups. (see Theorem 8.2 below) Suppose p > 1, M ⊂ Grp(R2p) is an affine chart, Ω ⊂ M is a bounded convex open subset of M, and there exists a discrete group Γ ≤ Aut(Ω) so that Γ\Ω is compact.
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