BIMSA >
BIMSA-Tsinghua Quantum Symmetry Seminar
Wreath-like product groups and their von Neumann algebras: W*-superrigidity and outer automorphism groups
Wreath-like product groups and their von Neumann algebras: W*-superrigidity and outer automorphism groups
Organizers
Speaker
Adrian Ioana
Time
Wednesday, March 5, 2025 10:30 AM - 12:00 PM
Venue
A3-3-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
Wreath-like products are a new class of groups, which are close relatives of the classical wreath products. Examples of wreath-like product groups arise from every non-elementary hyperbolic groups by taking suitable quotients. As a consequence, unlike classical wreath products, many wreath-like products have Kazhdan's property (T).
In this talk, I will focus on two main rigidity results for von Neumann algebras of wreath-like product groups obtained in joint work with Ionut Chifan, Denis Osin and Bin Sun. First, we show that any ICC group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group II1 factor L(G) remembers entirely the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T), confirming Connes’ rigidity conjecture from the early 1980s for these groups.
Second, for a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an ICC property (T) group G. This gives the first calculations of outer automorphism groups of II1 factors arising from property (T) groups, and can be viewed as a converse of Connes’ 1980 result showing that any such outer automorphism group is countable.
References:
arXiv:2111.04708 (Ann. of Math. 2023)
arXiv:2304.07457
arXiv:2402.19461
In this talk, I will focus on two main rigidity results for von Neumann algebras of wreath-like product groups obtained in joint work with Ionut Chifan, Denis Osin and Bin Sun. First, we show that any ICC group G in a natural family of wreath-like products with property (T) is W*-superrigid: the group II1 factor L(G) remembers entirely the isomorphism class of G. This provides the first examples of W*-superrigid groups with property (T), confirming Connes’ rigidity conjecture from the early 1980s for these groups.
Second, for a wider class wreath-like products with property (T), we show that any isomorphism of their group von Neumann algebras arises from an isomorphism of the groups. As an application, we prove that any countable group can be realized as the outer automorphism group of L(G), for an ICC property (T) group G. This gives the first calculations of outer automorphism groups of II1 factors arising from property (T) groups, and can be viewed as a converse of Connes’ 1980 result showing that any such outer automorphism group is countable.
References:
arXiv:2111.04708 (Ann. of Math. 2023)
arXiv:2304.07457
arXiv:2402.19461
Speaker Intro
Adrian Ioana is a Professor at the University of California, San Diego, acclaimed for his groundbreaking work in functional analysis, operator algebras, and ergodic theory, with a particular focus on von Neumann algebras and group actions. He was an invited speaker at the International Congress of Mathematicians in 2018.