Discrete subgroups of Lie groups

Introductory Talk

A novel witness to incoherence of SL(5,Z)

Abstract: Motivated by a question of Stover, we discuss an example of a Zariski-dense finitely generated subgroup of SL(5,Z) that is not finitely presented.

https://www.ihes.fr/~/douba/Incoherence.pdf

Strictly convex cusps with nilpotent holonomy

Abstract: In hyperbolic geometry, we define cusps via the thick-thin decomposition as the unbounded connected components of the thin part. A well-known fact about hyperbolic cusps is that their holonomy is virtually abelian, even though the Margulis lemma suggests it would be possible to imagine cusps with virtually nilpotent holonomy. Convex projective geometry is a generalization of hyperbolic geometry that is much more "flexible" : a convex projective manifold is modeled on some bounded convex and open subset of an affine chart of the real projective space. More regular example are modeled on strictly convex domains, whose boundary contains no segment. Hyperbolic geometry is an example of strictly convex geometry. A thick-thin decomposition theorem was proven by Cooper, Long and Tillmann for strictly convex geometries, which enables one to define strictly convex cusps as above. However, works of Crampon-Marquis and Ballas-Cooper-Leitner show that cusps that are too regular, or of maximal rank, are all hyperbolic cusps. So these can only have virtually abelian holonomy. Cooper later gave an example of a non-maximal rank strictly convex cusp of dimension 9 whose holonomy is the Heisenberg group, a nilpotent (and non virtually abelian) group. We further his construction by building strictly convex cusps with holonomy any finitely generated torsion-free nilpotent group. The first part of the talk is a brief explanation of what is known about strictly convex cusps, and an exposition of the results, while the second part is devoted to explaining how to build such cusps.

Reducible Suspensions of Fuchsian Representations 

Abstract: It is well-established that all deformations of Fuchsian representations of closed surface groups irreducibly embedded in higher rank special linear groups (that is, Hitchin representations) are Anosov. The same cannot be said for reducibly embedded Fuchsian representations. We will investigate this phenomenon by computing how far a generalization of these representations can be deformed (in specific ways) until they are no longer Anosov.

Flexing and branched bending

Abstract: For a hyperbolic n-manifold, bending along a totally geodesic hypersurface is a well-studied method of producing deformations of the original hyperbolic structure. However, there are instances where certain computations show that deformations of a hyperbolic structure exist, even in the absence of totally geodesic hypersurfaces to bend along. In a paper of Cooper, Long, and Thistlethwaite, a manifold is called "flexible" if it has this property. But what explains when a manifold is flexible? In this talk, we will explore one potential answer: a generalization of bending, where instead we bend along totally geodesic branched complexes.

Primitive-stable and Bowditch actions on hyperbolic spaces

Abstract: In this talk, we will introduce Bowditch representations of the free group of rank two (defined by Bowditch in 1998) along with primitive stable representations (defined by Minsky in 2010). Recently, Series on one hand, and Lee and Xu on an other hand, proved that Bowditch and primitive stable representations with value in $PSL(2,C)$ are equivalent. This result can be generalised to representations with value in the isometry group of an arbitrary Gromov hyperbolic space. Our proof in this context is independent. We will also give a similar result for the fundamental group of the four-punctured sphere.

https://arxiv.org/abs/2211.14546

Combination theorems for geometrically finite convergence groups

Abstract: Convergence group actions give a simple dynamical framework for understanding geometric properties of discrete groups of isometries of negatively curved metric spaces: when X is a negatively curved metric space, it is possible to "see" the geometry of a discrete isometric action on X by only considering the induced action by homeomorphisms on the ideal boundary of X. In the first part of this talk, I define geometrically finite group actions and explain how to understand them using this framework. In the second part of the talk, I show how to use this idea to prove combination theorems for geometrically finite groups. This is joint work with Alec Traaseth.

https://arxiv.org/abs/2305.08011