  In this note we draw a connection between noncommutative algebra and
geometric group theory. Specifically, we ask whether it is possible to bound
the sequence of codimensions for an associative PI-algebra using techniques
from geometric group theory. The classic and best known bound on codimension
growth was derived by finding a particularly nice spanning set for the
multilinear polynomials of degree n inside the free algebra. This spanning set
corresponds to permutations in the symmetric group which are so-called d-good,
where d is the degree of an identity satisfied by the algebra. The motivation
for our question comes from the fact that there is an obvious relationship
between the word metric on the symmetric group and the property of being
d-good. We answer in the affirmative, by finding a spanning set that
corresponds to permutations which are large with respect to the word metric. We
provide an explicit algorithm and formula for calculating the size of the
resulting bound, and demonstrate that it is asymptotically worse than the
classic one.
