# Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture

## TL;DR

This paper demonstrates that the Baum-Connes assembly map is quantitatively an isomorphism for certain lacunary hyperbolic groups, including many with property (T) and counterexamples with coefficients.

## Contribution

It identifies a class of groups where the Baum-Connes conjecture holds quantitatively, encompassing groups with large girth graphs and property (T).

## Key findings

- The Baum-Connes assembly map is a quantitative isomorphism for lacunary hyperbolic groups.
- Includes groups with property (T) and counterexamples with coefficients.
- Contains many groups with large girth graph sequences in their Cayley graphs.

## Abstract

We show that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and we explain how to see that this class contains many examples of groups that contain graph sequences of large girth inside their Cayley graphs and therefore do not have property (A). This includes the known counterexamples to the Baum-Connes conjecture with coefficients, as well as many other monster groups that have property (T).