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# Geometric representation theory: Convex projective geometry

Codimension-1 Flats in Convex Projective Geometry- Martin Bobb (Michigan)

Abstract: Convex projective manifolds generalize hyperbolic manifolds while allowing for some similarities to non-positively curved spaces, and some interesting deformation theory. In this lecture we will discuss the structure of codimension-1 flats in compact convex projective manifolds.

Rank one phenomena in convex projective geometry- Mitul Islam (Michigan)

Abstract: The goal of this talk is to develop analogies between rank one non-positive curvature/ CAT(0) and convex projective geometry. We will introduce the notion of rank one automorphisms of properly convex domains and characterize them as contracting group elements. We will prove that a discrete rank one automorphism group is either virtually cyclic or acylindrically hyperbolic. This leads to some applications like computation of space of quasimorphisms, counting of closed geodesics, and genericity results.

Moduli space of unmarked convex projective surfaces- Zhe Sun (Luxembourg)

Abstract: Mirzakhani found a beautiful recursive formula to compute the volume of the moduli space of Riemann surfaces. We discuss the possible similar recursive formula where the Riemann surfaces are replaced by the convex projective surfaces. We investigate the boundedness of projective invariants, area, and many other notions that are uniformly related to each other and we show one of these bounded subsets has polynomially bounded Goldman symplectic volume.

## 10 replies on “Geometric representation theory: Convex projective geometry”

[…] Convex projective geometry: […]

Arielle_Leitnersays:

Awesome talk Martin! What can you use the tori for? (As in what are the applications of your theorem?)

Martin Bobbsays:

Great question, I didn’t get to say! The tori are codimension-1, so it makes sense to cut along them. This gives a decomposition, and Benoist showed in dimension 3 that the resulting collection of 3-manifolds are homeomorphic to finite-volume cusped hyperbolic manifolds (this is a JSJ decomposition, actually!). That comes from geometrization in dimension 3.
In general, the pieces need not have hyperbolic structures, but the right thing to call them is “cusped convex projective manifolds with type d cusps” where d is the dimension. Cusped convex projective manifolds were introduced by Cooper, Long, and Tillmann, and further studied and classified by Ballas, Cooper, and Leitner (though I think you knew that part).
I would say this theorem means that if you want to understand compact convex projective manifolds that have codimension-1 (virtual) tori, you need to understand cusped convex projective manifolds (with type d cusps at least).

ZSunsays:

Great talk, Martin! I’m just wondering if you could put the invariants on the codim 1 tori using the construction in “Tetrahedra of flags, volume and homology of SL(3), arXiv:1101.2742”, then possibly implies some topological consequences. Also, I guess the Hilbert volume of 3 manifold is not clear yet?

Martin Bobbsays:

Mitul, really great stuff! I’m happy about the rank-rigidity results.

mitul_islamsays:

Hi Martin, thank you so much! It would be very interesting to find out more about rank-rigidity in the convex co-compact case.

Martin Bobbsays:

Hey Zhe, it’s interesting to see the triangle triple ratio show up as a correction term (as long as I’m understanding the Fermi-Dirac integral…).

ZSunsays:

Hi Martin, thank you for your comment. Given a convex projective structure, the triple ratio is a function of ordered distinct 3 points on a circle which is invariant under pi1(S). {ordered distinct 3 points }/p1(S)=T1(S). Thus it is a function of the unit tangent bundle of the surface which is compact. Thus the triple ratios are bounded above and below. You could think the triple ratio is some kind of a correction term.

Arielle_Leitnersays:

Great talk Mitul! Could you give more motivation for why you called them rank one automorphisms? How does it connect to the rank of a symmetric space in the way I am used to thinking about it? Thanks!

mitul_islamsays:

Hi Arielle, thank you very. much! That is an excellent question.

The motivation for the name rank one indeed originates in the symmetric space setting. In a Riemannian symmetric space, we can think of rank as the dimension of the maximal abelian semi-simple subalgebra which geometrically corresponds to maximal totally geodesic flats. But in order to generalize this notion to Riemannian manifolds, one asks for an infinitesimal version of this condition (since demanding totally geodesic flats could be too restrictive). That condition is in terms of the dimension of the space of parallel Jacobi fields (called “rank” or “Euclidean rank”) for the axis of an isometry. The ‘negative-curvature like’ isometries have this rank (in Jacobi field sense)=1, hence the name rank one. So, the connection with the usual rank would be the following: rank one is an infinitesimal analog of being contained in a totally geodesic flat of dimension at most 1.

This Jacobi field condition turns into a half flat condition in CAT(0), motivating my half triangle definition.

There are more analogies with Riemannian non-positive curvature motivating this: Riem. rank 1 isometries have negative curvature’ properties, like Morse lemma for their axis, the contraction property of projections onto the axis, etc. The projective ‘rank one’ automorphisms also have similar properties – contracting projections and Morse lemma (I try to give some outline of this in Part II of my talk).

So the name rank one is motivated by this (weak) analogy between classical non-pos. curvature and convex projective geometry – the philosophy being that any ‘negative curvature’ like behaviour is going to be concentrated entirely on isometries that show rank one’ behaviour.