Uniform amenability at infinity

TL;DR

The paper introduces uniform amenability at infinity for groups, demonstrating it for many groups including free groups, and shows it leads to strong convergence properties in operator algebras.

Contribution

It defines uniform exactness for groups and proves it for a broad class, revealing new convergence phenomena in operator algebra contexts.

Findings

Uniform amenability at infinity holds for free groups and their limits.

Convergent sequences of such groups exhibit strong operator algebraic convergence.

Spectral radius formulas converge uniformly over measures with fixed support size.

Abstract

We introduce the notion of uniform exactness, or uniform amenability at infinity, for discrete groups and prove it for a wide class of groups containing free groups and their limit groups. This shows a novel strong convergence phenomenon that any convergent sequence of such groups in the space of marked groups converges strongly in the operator algebraic sense. In particular, convergence of the spectral radius formula is uniform over probability measures on such groups whose supports have a fixed cardinality.