  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.
