| Recently, there has been fruitful interaction between construction of free subgroups of algebraic groups and areas of mathematics surprisingly far afield from algebraic groups. The main technique for producing free subgroups is dynamical: Roughly speaking, one wants to find an action of G wherein a prescribed element and its inverse have attracting points with large basins of attraction, and the rate of attraction is somewhat uniform.;Tits' alternative has it that if the Zariski closure of a finitely generated linear group Gamma is semisimple, then it Gamma contains a nonabelian free subgroup. Let G be a connected, semisimple algebraic group, Gamma a center-free, Zariski-dense subgroup. We show (under additional hypotheses about the way that Gamma is allowed to meet certain simple factors of G) that for any finite set F of non-identity elements of Gamma, there exists an element gamma ∈ Gamma of infinite order, which satisfies no nontrivial relation with any element of Gamma. In fact, we show that for any such F, it is easy to find such gamma: the set of such gamma is dense in the profinite completion of Gamma.;As an application, we note that if Gamma as above is countable, then the reduced C*-algebra Cr* (Gamma) of Gamma is simple and has a unique trace up to normalization, answering (most of) a question of Bekka and de la Harpe, and generalizing earlier results of Bekka, Cowlings, and de la Harpe. |
