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Random quotients of acylindrically hyperbolic groups
Abstract: Quotients of hyperbolic groups and their generalizations have long been a powerful tool for constructing groups with interesting algebraic properties. In this talk, I will describe what happens to the geometry of a group when we impose random relations, obtained via random walks. I will focus on acylindrically hyperbolic groups, a class that includes non-elementary hyperbolic groups, mapping class groups of most finite-type surfaces, and Out(F_n) for n at least 2. Under suitable hypotheses on the random walks, these random quotients are asymptotically almost surely again acylindrically hyperbolic. I will explain the geometric ideas behind this result and why random quotients also preserve hyperbolicity and relative hyperbolicity. This is joint work with Dan Berlyne, Giorgio Mangioni, Thomas Ng, and Alexander Rasmussen.
Universal algebra and limiting densities
Abstract: We consider the question of what it means to be a typical structure in an algebraic variety and define this concept formally using limiting densities. We then discuss several possible situations, including that in which the sentences true in the typical structure are exactly those true in the free structure, that in which each sentence has limiting density 0 or 1 but the theory of the typical structure may not be that of the free structure, and that in which there is a sentence with limiting density strictly between 0 and 1.
This work is joint with Meng-Che "Turbo" Ho and Julia Knight.
Spectral theory and descriptive combinatorics for Borel pmp graphs
Abstract: In classical spectral graph theory, one associates to finite graphs Hermitian matrices, such as the adjacency matrix, that encode graphical information. The spectral properties of these matrices are then leveraged to analyze the combinatorial features, including vertex coloring, edge coloring, and matching, of the corresponding graphs. Analogously, one can associate bounded, self-adjoint operators, such as the adjacency operator, to Borel pmp graphs of bounded degree. In this talk, we present new descriptive combinatorics results for pmp graphs that involve the application of spectral theory to their associated operators. This is joint work with Pieter Spaas, Alexander Tenenbaum, and Sofia Torelli.
Nonhyperfiniteness in Invariant Percolation
Abstract: Every invariant percolation process has an associated measurable graph called a cluster graphing. This connection establishes parallels between some behaviours of the random process and properties of the measured graph. In this talk, I will outline how transporting the notion of nonhyperfinitness from the study of measured equivalence relations allows us to prove results about invariant percolation on nonunimodular graphs. Results in this talk are joint with Sasha Bell, Tasmin Chu, Greg Terlov, and Anush Tserunyan; and with Gábor Pete, Greg Terlov, and Ádám Timár.
Measurable Transformations on the p-adics: Weak Mixing Examples
Abstract: Transformations on the p-adic numbers provide an interesting source of examples of measure-preserving transformations. We will briefly survey work on ergodic isometries on the p-adics and also mixing examples. Then we will consider constructions adapted to the p-adics that are rank-one and yield weakly mixing examples, and discuss when they are continuous. This is joint work with Joanna Furno and Rauan Kaldybaev.
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