"Product set growth and groups acting on trees"Markus Oliver SteenbockIn the theory of infinite groups, it is a classical topic to study the growth rates of balls in the Cayley graph of a finitely generated group. On the one hand, this does not depend on the choice of the generating set of the group. On the other hand, it can be used as a geometric invariant. For instance, by Gromov's polynomial growth theorem, virtually nilpotent groups are exactly the groups with a polynomial growth rate. In contrast, the non-abelian free group of rank k acts by isometries transitively and freely on its Cayley graph, a tree of vertex degree = 2k. In this case the growth rate of the non-abelian free group corresponds to the growth of balls in the tree. These have exponential growth. In this talk, we discuss fine estimates on the growth of finite subset of non-abelian free groups, or more generally, of groups acting on trees. In other words, we are interested in fine estimates - as opposed to its asymptotic behaviour - of the growth of free sub-semigroups. To this point, we discuss an improvement of a result of Razborov by Safin that states that there is a uniform constant b>0 such that for any subset U of a non-abelian free group, |U^3|> b |U|^2 unless U is a subset of an infinite cyclic subgroup. |
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