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Does $SL(2,mathbbZ)$ have a finite-dimensional faithful unitary representation? No such representation exists for $SL(2,mathbbR)$, but I don't see a reason why one shouldn't exist for $SL(2,mathbbZ)$.

Here a non-explicit proof of the existence of a faithful representation of $mathrmSL_2(mathbfZ)$ in $mathrmSU(2)$, using basic algebraic geometry and topology, and relying on the amalgam decomposition of $mathrmSL_2(mathbfZ)$.

[The basic idea is that if all representations in $mathrmSU(2)$ were non-faithful, by Zariski-density this would also be the case for representations into $mathrmSL_2$. We need to use the particular form of the presentation of $mathrmSL_2(mathbfZ)$, since the argument will not carry over representations of $mathrmSL_3(mathbfZ)$ in $mathrmSU(3)$.]

Let $P_t$ be the set of $2times 2$ matrices with determinant 1 and trace $t$. Both $P_0$ and $P_1$ are irreducible as algebraic varieties (being $mathrmSL_2$ conjugacy classes). Then for $K$ a field of characteristic zero, $P_0(K)$ is the set of elements of order 4 in $mathrmSL_2(K)$, and $P_1(K)$ is the set of elements of order 6 in $mathrmSL_2(K)$.

For every $(g,h)in P_0times P_1$, $g^2=h^3$ equals $-I_2$. Hence the set of representations of $$mathrmSL_2(mathbfZ)=langle u,vmid u^4=v^6=[u^2,v]=[u,v^3]=1rangle$$ (restricting to those for which the image of $u$ has order 4 and the order of $v$ has order 6)
into $mathrmSL_2(K)$ can be naturally identified to $(P_0times P_1)(K)$. Note that $P_0times P_1$ is irreducible.

Write $P_t^sharp=P_t(mathbfC)capmathrmSU(2)$. Then using that $mathrmSU(2)$ is Zariski-dense in $mathrmSL_2(mathbfC)$ and describing $P_t(mathbfC)$ as a conjugacy class, one deduces that $P_t^sharp$ is Zariski-dense in $P_t(mathbfC)$. So $P_0^sharptimes P_1^sharp$ (which is a 4-dimensional real manifold) is Zariski-dense in $(P_0times P_1)(mathbfC)$.

For every given nontrivial element $w$ in $mathrmSL_2(mathbfZ)$ the set of representations vanishing on $w$ is a proper subvariety of $P_0times P_1$, hence has dimension $le 3$. By Zariski density of $P_0^sharptimes P_1^sharp$, we deduce that its intersection with $P_0^sharptimes P_1^sharp$ is a proper Zariski closed subset (because $mathrmSL_2(mathbfZ)$ admits one faithful representation into $mathrmSL_2(mathbfC)$, the standard one); in particular it has empty interior (in the ordinary topology). By the Baire theorem, the union over all $w$ is still a proper subset. Hence we deduce the existence of an element of $P_0^sharptimes P_1^sharp$ defining a faithful representation.