November 2020 – Midwest dynamics and group actions seminar

11/30: Stephen Cantrell

Counting limit theorems for dominated representations

Abstract: In this talk we will discuss how to use symbolic dynamics and thermodynamic formalism to study dominated (or Anosov) representations. These are certain representations of finitely generated groups into general linear groups that can be viewed as higher rank analogies of cocompact, isometric group actions on the hyperbolic plane. We will show how to use our dynamical view point to study the statistics of random matrix products in this setting. This talk is based on joint work with Rhiannon Dougall, Italo Cipriano and Cagri Sert.

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11/23: Andreas Wieser

Simultaneous supersingular reductions of CM elliptic curves

Abstract: In joint work with Menny Aka, Manuel Luethi and Philippe Michel, we study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show – under additional congruence assumptions – that the reductions are surjective on the product of supersingular loci when the discriminant of the order becomes large. The goal for this talk is to explain this result while emphasizing the role of homogeneous dynamics.

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11/16: Jonathan DeWitt

Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds

Abstract: Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,\dots ,R_m$ generate the isometry group of $M$ . Let $f_1,\dots ,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from $S^n$ to real, complex, and quaternionic projective spaces.

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11/09: Andrew Zimmer

Convex co-compact representations of 3-manifold groups

Abstract: A representation of a finitely generated group into the projective linear group is called convex co-compact if it has finite kernel and its image acts convex co-compactly on a properly convex domain in real projective space. In this talk I will discuss the case of 3-manifold groups and prove that the fundamental group of a closed irreducible orientable 3-manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic, or Euclidean × Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. This extends a result of Benoist about convex real projective structures on closed 3-manifolds. In each case, I will also describe what these representations look like. This is joint work with Mitul Islam (a graduate student at the University of Michigan).

Slides