Random groups and Property (T): \.Zuk's theorem revisited

TL;DR

This paper rigorously proves that random groups in the Gromov density model for d > 1/3 almost surely have property (T), fixing gaps in Zuk's original proof using combinatorial and spectral methods.

Contribution

It provides a complete, rigorous proof of Zuk's theorem on property (T) for random groups, including alternative approaches and fixing previous gaps.

Findings

Random groups in the Gromov density model for d > 1/3 have property (T) with high probability.

The proof uses combinatorial arguments and spectral properties of random graphs.

An alternative proof avoids combinatorial complexities by relying on spectral graph theory.

Abstract

We provide a full and rigorous proof of a theorem attributed to \.Zuk, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability. The original paper had numerous gaps, in particular, crucial steps involving passing between different models of random groups were not described. We fix the gaps using combinatorial arguments and a recent result concerning perfect matchings in random hypergraphs. We also provide an alternative proof, avoiding combinatorial difficulties and relying solely on spectral properties of random graphs in G(n, p) model.