A theorem of Bestvina-Bromberg-Fujiwara asserts that the mapping class group of a hyperbolic surface of finite type has finite asymptotic dimension; its proof relies on an earlier result of Bell-Fujiwara stating that the curve complex has finite asymptotic dimension. The analogous statements are still open for Out(Fn). In joint work with Mladen Bestvina and Ric Wade, we give a first hint towards this, by obtaining a bound (linear in the rank n) on the topological dimension of the Gromov boundary of the graph of free factors of Fn (as well as some other hyperbolic Out(Fn)-graphs).

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