  We show that the virtual cohomological dimension of a Coxeter group is
essentially the regularity of the Stanley--Reisner ring of its nerve. Using
this connection between geometric group theory and commutative algebra, as well
as techniques from the theory of hyperbolic Coxeter groups, we study the
behavior of the Castelnuovo--Mumford regularity of square-free quadratic
monomial ideals. We construct examples of such ideals which exhibit arbitrarily
high regularity after linear syzygies for arbitrarily many steps. We give a
doubly logarithmic bound on the regularity as a function of the number of
variables if these ideals are Cohen--Macaulay.
