Let G be a finitely generated group acting faithfully by linear transformations on a finite-dimensional complex vector space. The theorems of Malcev, Selberg, or Tits provide important properties satisfied by G. To what extent do these properties continue to hold when G is acting by polynomial (instead of linear) transformations? In order to address this question, I shall describe a few results that illustrate how one can use p-adic or finite fields for problems which are initially phrased in terms of complex numbers.
Fanny Kassel