# Certifying numerical estimates of spectral gaps

## TL;DR

This paper introduces a method using conic optimization to certify lower bounds on spectral gaps of Laplace operators on groups, providing a constructive proof of Kazhdan property (T) and software tools for broader applications.

## Contribution

It presents a novel conic optimization approach to certify spectral gaps and Kazhdan property (T) for groups, with software for general finitely presented groups.

## Key findings

- Lower bounds on spectral gaps established for specific groups.
- Constructive proof of Kazhdan property (T) using optimization.
- Software implementation for spectral gap certification.

## Abstract

We establish a lower bound on the spectral gap of the Laplace operator on special linear groups using conic optimisation. In particular, this provides a constructive (but computer assisted) proof that these groups have Kazhdan property (T). A software for such optimisation for other finitely presented groups is provided.