October 2020 – Midwest dynamics and group actions seminar

11/02: Minju Lee

Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Abstract: This is joint work with Hee Oh. We establish an analogue of Ratner’s orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$ , assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with Fuchsian ends. For $d = 3$ , this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any $k\geq 1$ ,
(1) the closure of any $k$ -horosphere in $M$ is a properly immersed submanifold;
(2) the closure of any geodesic $(k+1)$ -plane in $M$ is a properly immersed submanifold;
(3) an infinite sequence of maximal properly immersed geodesic $(k+1)$ -planes intersecting $\mathrm{core} M$ becomes dense in $M$ .

Slides

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10/26: Homin Lee

Global rigidity theorems for actions of higher rank lattices

Abstract: We will talk about rigidity theorems for smooth actions of a higher rank lattice $\Gamma$ on compact manifolds following the philosophy of the Zimmer program. Previously many global rigidity phenomena are known due to the presence of “higher rank” with “property (T)” on the lattice side, and “Anosov” on the dynamics side. In this talk, we will discuss two global rigidity theorems that relax each condition. The main ingredients are cocycle superrigidity and its generalizations.

Slides

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10/19: Florian Richter

Additive and geometric transversality of fractal sets in the reals and integers

Abstract: Using the language of fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the 1960s that explore the relationship between digit expansions of real numbers in distinct prime bases. While his famous x2 x3 conjecture remains open, recent solutions to some of his “transversality conjectures” have shed new light on old problems. In this talk we explore analogues of results surrounding Furstenberg’s conjectures in the discrete setting of the integers, with the aim of understanding the independence of sets of integers that are structured with respect to different prime bases. This is based on joint work with Daniel Glasscock and Joel Moreira.

Slides

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10/12: Ben Lowe

Minimal Surfaces in Negatively Curved 3-Manifolds and Dynamics

Abstract: The Grassmann bundle of tangent 2-planes over a closed hyperbolic 3-manifold M has a natural foliation by (lifts of) immersed totally geodesic planes in M. I am going to talk about work I’ve done on constructing foliations whose leaves are (lifts of) minimal surfaces in a metric on M of negative sectional curvature, which are deformations of the totally geodesic foliation described above. The foliations we construct make it possible to use homogeneous dynamics to study how closed minimal surfaces in variable negative curvature are distributed in the ambient 3-manifold. Many of the ideas here come from recent work of Calegari-Marques-Neves. I was able to prove some preliminary results on the dynamics of these foliations, but much remains to be understood.

Slides