# Subspace Arrangements and Property T

## TL;DR

This paper improves the geometric method for establishing Kazhdan property T in groups generated by finite subgroups, providing better bounds and applying the results to finite Coxeter groups for spectral gap analysis.

## Contribution

It extends the geometric approach to property T, offering improved bounds and explicit Kazhdan constants for finite Coxeter groups.

## Key findings

- Enhanced bounds for property T criteria.
- Explicit Kazhdan constants for finite Coxeter groups.
- Generalization of spectral gap results for random walks.

## Abstract

We reformulate and extend the geometric method for proving Kazhdan property T developed by Dymara and Januszkiewicz and used by Ershov and Jaikin. The main result says that a group G, generated by finite subgroups G_i, has property T if the group generated by each pair of subgroups has property T and sufficiently large Kazhdan constant. Essentially, the same result was proven by Dymara and Januszkiewicz, however our bound for "sufficiently large" is significantly better. As an application of this result, we give exact bounds for the Kazhdan constants and the spectral gaps of the random walks on any finite Coxeter group with respect to the standard generating set, which generalizes a result of Bacher and de la Hapre.