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

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.