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Is there an example of an injective homomorphism $\pi: F_2\to U(n)$ of the 2 generator free group $F_2$ in some unitary group of matrices $U(n)$?

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Yes, of course. This is one of the main points of Banach-Tarski. Explicit constructions can be found here, for example:… – Theo Buehler Jun 30 '11 at 22:47
Dear Theo, did i missed something or the examples in these explicit constructions the generators are invertible, but not unitary matrices? many thanks – Paulo Jun 30 '11 at 23:25
@Paulo: the constructions there include an injective homomorphism into $\text{SO}(3)$, which is compact Lie, so embeds in a unitary group (in this case we can take $\text{U}(3)$). – Qiaochu Yuan Jun 30 '11 at 23:27
Using the fact that the universal cover of $SO(3)$ is $SU(2)$, together with the fact that every homomorphism of the free group can be lifted (a consequence of freeness), you can construct explicit homomorphisms of $F_2$ into $SU(2)$. – Alain Valette Jul 1 '11 at 12:59
Of course one could invoke the Tits alternative, stating that in characteristic 0 any not virtually solvable subgroup of $GL_n$ contains $F_2$... but that would be a bit pedantic! – Alain Valette Jul 2 '11 at 19:46

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