A deep result of Furstenberg from 1967 states that if Γ is a lattice in a semisimple Lie group G, then there exists a measure on Γ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of G is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.
Let SO3(ℝ) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure μ. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO3(ℝ) with sufficiently small measure, then μ(A2) > 3.99 μ(A).
The Zimmer programme asks how lattices in higher-rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SLn(ℝ) with n ≥ 3 or other higher rank ℝ-split semisimple Lie groups, the action is trivial up to a finite group action. In this talk, we will explain what happens in the critical dimension for higher rank ℝ-split semisimple Lie groups. For example, non-trivial actions by lattices in SLn(ℝ), n ≥ 3, on (n-1)-dimensional manifolds are isomorphic to the standard action on ℝPn-1 up to a finite quotient group and a finite covering.
The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| ≫ min ( |A|1+c, |G| ). In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact simple Lie group G, then μ(AAA) > min ( (3+c)μ(A), 1 ), where μ is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.
You must be logged in to post a comment.