A note of Zuk's criterion

TL;DR

This paper discusses Zuk's criterion for Property(T) in finitely generated groups, providing examples that demonstrate the criterion's optimality and cannot be improved.

Contribution

The paper presents two examples showing that Zuk's criterion for Property(T) cannot be strengthened, confirming its optimality.

Findings

Zuk's criterion is optimal for Property(T)

Two specific examples demonstrate the criterion's limits

The smallest non-zero eigenvalue condition is sharp

Abstract

Zuk's criterion give us a condition for a finitely generated group to have Property(T): the smallest non - zero eigenvalue of Laplace operator corresponding to the simple random walk on the associated graph have to be greater than 1/2. We present here two examples that prove that this condition cannot be improved.