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A bound on the codimensions of a PI-algebra using group geometry

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.