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We prove two results about the width of words in
. The first is that, for every
, there is a constant
such that the width of any word in
is less than
. The second result is that, for any word
is big enough, the width of
is at most 87.
We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GLd have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.
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