Groups and dynamics seminar

Time: Thursday, July 3, 10:00
Ross 70

 Title: On unimodular random hyperbolic manifolds


 Abstract:
 We will discuss unimodular random hyperbolic manifolds. Our goal is to relate their topology and their dynamical properties. More concretely, it was shown by Biringer-Raimbault that unimodular random manifolds have 0,1,2 or a Cantor space worth of topological ends. We show how the end structure controls properties such as recurrence vs transience, the Liouville property and positivity of the Cheeger constant, extending the familiar picture for normal covers. The main tools include Poisson point processes, Delaunay triangulations and the mass transport principle. The talk is based on joint work in progress with Ilya Gekhtman, Nir Lazarovich and Asaf Nachmias.
Around 2013, a natural conjecture by Julia Knight anticipated that given a first order sentence, its truth value over a random group in the few relators model, is equivalent to its truth value over non-abelian free groups.
Our work extends the framework of this conjecture to random groups in the Gromov Density Model, at optimal density d<1/2. 
We prove the conjecture for sentences that belong to the Boolean algebra of universal sentences, as well as for sentences of minimal rank with arbitrary number of quantifiers, for random groups of density d<1/2.
In this talk, we will present our results and, as time permits, outline the key strategies and ideas involved in the proofs.
Note: Sobhi's lecture is part of the requirements to complete his thesis. It will largely overlap with the colloquium he gave last year.