  Geometric grid classes of permutations have proven to be key in
investigations of classical permutation pattern classes. By considering the
representation of gridded permutations as words in a trace monoid, we prove
that every geometric grid class has a growth rate which is given by the square
of the largest root of the matching polynomial of a related graph. As a
consequence, we characterise the set of growth rates of geometric grid classes
in terms of the spectral radii of trees, explore the influence of "cycle
parity" on the growth rate, compare the growth rates of geometric grid classes
against those of the corresponding monotone grid classes, and present new
results concerning the effect of edge subdivision on the largest root of the
matching polynomial.
