| Résume |
Let G be an ind-definable group acting faithfully on a definable
set X in the complex field. We are interested in the following question:
are there a definable subset D of G, and arbitrarily large finite sets
A_n in D and B_n in X, such that |A_n*B_n|<= |B_n|^{1+epsilon} for every
epsilon>0 and all sufficiently large n? Under suitable non-degeneracy
assumptions, we expect such non-expansion phenomenon to occur only when the
sets A_n are concentrated on a coset of a nilpotent algebraic subgroup
of G.
In this talk we will discuss two specific examples, the first is the group
of polynomial automorphisms of the complex plane C^2. The second is the
multiplicative group of the function field C(t), acting on C^2 via
f(t)*(x,y):=(x, f(x)y). This is joint work in progress with Martin Bays.
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