On Garland's vanishing theorem for $\mathrm{SL}_n$

TL;DR

This paper explains Garland's vanishing theorem for $ ext{SL}_n$, highlighting its proof method linking cohomology vanishing to eigenvalues of combinatorial Laplacians, with broad applications in mathematics.

Contribution

It provides an accessible exposition of Garland's theorem for $ ext{SL}_n$, emphasizing the proof technique and its significance across multiple mathematical fields.

Findings

Cohomology groups vanish under certain conditions.

Eigenvalues of combinatorial Laplacians are crucial for cohomology vanishing.

The theorem has diverse applications in representation theory, group theory, and combinatorics.

Abstract

This is an expository paper on Garland's vanishing theorem specialized to the case when the linear algebraic group is $\mathrm{SL}_n$. Garland's theorem can be stated as a vanishing of the cohomology groups of certain finite simplicial complexes. The method of the proof is quite interesting on its own. It relates the vanishing of cohomology to the assertion that the minimal positive eigenvalue of a certain combinatorial laplacian is sufficiently large. Since the 1970's, this idea has found applications in a variety of problems in representation theory, group theory, and combinatorics, so the paper might be of interest to a wide audience. The paper is intended for non-specialists and graduate students.