Organizers


 

RTG WORKSHOPS & LECTURE SERIES

Random Walks on Groups

March 29 - 31, 2013

Room B 844, East Hall
University of Michigan
Ann Arbor, MI

Schedule

Friday

1:00 - 2:00 pm Vadim Kaimanovich
Random walks and Poisson boundaries; entropy and the Liouville property

Abstract: We shall give a general introduction to the boundary theory of random walks on groups starting from general notions and illustrating them on concrete examples of groups with hyperbolic properties and self-similar groups

 

2:15 - 3:15 pm Anders Karlsson
An ergodic theorem for noncommuting products

Abstract: In this introductory lecture I will recall some classical ergodic theorems (Birkhoff, Kingman, Oseledets) and formulate a rather general ergodic theorem for noncommuting products that appeared in my joint work with F. Ledrappier. This involves the notion of horofunctions which I will spend some time explaining.

 

4:00 - 5:00 pm Alex Furman
Products of random matrices: Lyapunov exponents and stationary measures

Abstract: In these three talks we shall focus on asymptotic characteristics of
products of random i.i.d. matrices, that can be viewed as random walks on matrix groups. Writing the random product in the polar form (KAK decomposition) the behavior of the A-component is described by the Lyapunov exponents, and the distribution of the K-part by the stationary measure. These notions will be introduced in the first lecture.

6:00 pm Banquet at Isalita

 

Saturday

9:00 - 10:00 am Vadim Kaimanovich
Boundary convergence and identification; applications to the mapping class group

 

10:15 - 11:00 am Giulio Tiozzo
Geodesic ray tracking for random walks on groups

Abstract: Given a finitely generated group G acting on a geodesic space X and a probability measure
on G, one can construct a random walk by choosing at each step a random group element and letting it act on X. The natural question arises whether the sample paths can be approximated by some geodesic in X. We will prove that, in a quite general setting, the sample path and the limiting geodesic lie within sublinear distance. Our argument applies to the case of the mapping class group acting on Teichmueller space, answering a question of Kaimanovich. Another application includes the statistics of excursions of random Teichmueller geodesics in the thin part of moduli space.

 

11:15 -12:15 am Alex Furman
Lyapunov exponents: positivity of the top exponent, simplicity of the spectrum, regularity

Abstract: The second lecture will examine qualitative behavior of the Lyapunov exponents,
and some regularity results about Lyapunov exponents and stationary measures.

1:30 - 2:30 pm Moon Duchin
Random Teichmüller geodesics

Abstract: I'll review some of the features and pathologies of geodesics in the Teichmüller metric, including the phenomena that are obstructions to hyperbolicity. Then I'll discuss recent work with Dowdall and Masur in which we work out properties enjoyed by generic geodesics, concluding that these obstructions are quantifiably rare, and Teichmüller space is in this sense "statistically hyperbolic." One key tool, devised by Eskin and Mirzakhani, shows that geodesics are well-modeled by random walks on a net of points.

 

2:45 - 3:30 pm Andrew Zimmer
The Poisson and Martin boundary of a harmonic manifold

Abstract: A complete Riemannian manifold is called harmonic if each geodesic sphere of sufficiently small radii has constant mean curvature. Examples of harmonic manifolds include flat spaces and rank one locally symmetric spaces. The Lichnerowicz conjecture asks if these are the only compact harmonic manifolds. In this talk we will present some evidence that this is the case. In particular, we will discuss various compactifications of non-compact non-flat simply connected harmonic manifolds. We will show that the Martin, Poisson, Busemann, and "geometric" boundaries coincide. Moreover, in this case, the harmonic measure can be identified with "visual" measure. This leads to several corollaries concerning the fundamental group of a compact harmonic manifold and the dynamics of the geodesic flow.

 

4:00 - 5:00 pm Anders Karlsson
An ergodic theorem for noncommuting products II

Abstract: In the second talk I will give a proof of the noncommutative ergodic theorem and explain a few of its consequences, notably to the drift of random walks on finitely generated groups and the existence of non-constant bounded harmonic functions improving on a result of Varopoulos.

 

Sunday

9:00 - 10:00 am Vadim Kaimanovich
Random walks and amenability; applications to self-similar groups

 

10:30 - 11:30 am Anders Karlsson
An ergodic theorem for noncommuting products III

Abstract: In the final lecture, I hope to describe some further concrete cases of the main theorem, such as Oseledets' multiplicative ergodic theorem and a random version of Thurston's spectral theorem for surface homeomorphisms.

 

12:00 -1:00 pm Alex Furman
An application to product actions on manifolds

In the third lecture we shall see an application of (a cocycle version) of some of the above ideas to a problem that does not involve random walks, namely constraints on actions of certain groups on compact manifolds.

 

 


Sponsors
National Science Foundation,
Michigan Mathematical Journal,
Department of Mathematics at the University of Michigan

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