Consequences of the random matrix solution to the Peterson-Thom conjecture

TL;DR

This paper leverages recent solutions to the Peterson-Thom conjecture to reveal new structural properties of free group factors, advancing understanding of their subalgebra configurations and embedding behaviors.

Contribution

It introduces several new structural results for free group factors, including resolutions to longstanding conjectures and generalizations of key solidity theorems.

Findings

Resolution of the coarseness conjecture.

Generalized strong solidity results.

Dichotomy for maximal amenable subalgebras.

Abstract

In this paper we show various new structural properties of free group factors using the recent resolution (due independently to Belinschi-Capitaine and Bordenave-Collins) of the Peterson-Thom conjecture. These results include the resolution to the coarseness conjecture independently due to the first-named author and Popa, a generalization of Ozawa-Popa's celebrated strong solidity result using vastly more general versions of the normalizer (and in an ultraproduct setting), a dichotomy result for intertwining of maximal amenable subalgebras of interpolated free group factors, as well as application to ultraproduct embeddings of nonamenable subalgebras of interpolated free group factors.