Strong convergence of unitary and permutation representations of discrete groups
Michael Magee
TL;DR
This survey develops and clarifies the framework of strong convergence for finite-dimensional unitary and permutation representations of discrete groups, introducing $\,\mathrm{P}\mathbf{Mat}\mathrm{F}$ and $\mathrm{P}\mathbf{Perm}\mathrm{F}$ as robust approximation notions to the regular representation. It surveys foundational properties, inheritance principles, and the interaction with Fell topology, while detailing both positive results (e.g., for amenable groups, free groups, limit groups, and RAAGs) and non-examples (notably certain $\mathrm{SL}_d(\mathbb{Z})$ cases). A central theme is the induction principle: strong convergence is preserved under induction from cocompact lattices to ambient groups, enabling spectral-convergence-type statements to Plancherel measures and applications to geometry. In particular, for hyperbolic surfaces, random permutation representations and random coverings yield convergence of spectral data and Laplacian spectra, linking representation-theoretic convergence to geometric spectral behavior on families of surfaces. The work highlights both the deep connections to random matrix and random graph theory and the potential for extending these methods to broader geometric and spectral problems.
Abstract
We survey a research program on the strong convergence of unitary and permutation representations of discrete groups. We also take the opportunity to flesh out details that have not appeared elsewhere.