Finite-dimensional faithful unitary representations of SL(2,Z)

Question

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

Answer 1

Here a non-explicit proof of the existence of a faithful representation of $\mathrm{SL}_2(\mathbf{Z})$ in $\mathrm{SU}(2)$, using basic algebraic geometry and topology, and relying on the amalgam decomposition of $\mathrm{SL}_2(\mathbf{Z})$.

[The basic idea is that if all representations in $\mathrm{SU}(2)$ were non-faithful, by Zariski-density this would also be the case for representations into $\mathrm{SL}_2$. We need to use the particular form of the presentation of $\mathrm{SL}_2(\mathbf{Z})$, since the argument will not carry over representations of $\mathrm{SL}_3(\mathbf{Z})$ in $\mathrm{SU}(3)$.]

Let $P_t$ be the set of $2\times 2$ matrices with determinant 1 and trace $t$. Both $P_0$ and $P_1$ are irreducible as algebraic varieties (being $\mathrm{SL}_2$ conjugacy classes). Then for $K$ a field of characteristic zero, $P_0(K)$ is the set of elements of order 4 in $\mathrm{SL}_2(K)$, and $P_1(K)$ is the set of elements of order 6 in $\mathrm{SL}_2(K)$.

For every $(g,h)\in P_0\times P_1$, $g^2=h^3$ equals $-I_2$. Hence the set of representations of $$\mathrm{SL}_2(\mathbf{Z})=\langle u,v\mid u^4=v^6=[u^2,v]=[u,v^3]=1\rangle$$ (restricting to those for which the image of $u$ has order 4 and the order of $v$ has order 6) into $\mathrm{SL}_2(K)$ can be naturally identified to $(P_0\times P_1)(K)$. Note that $P_0\times P_1$ is irreducible.

Write $P_t^\sharp=P_t(\mathbf{C})\cap\mathrm{SU}(2)$. Then using that $\mathrm{SU}(2)$ is Zariski-dense in $\mathrm{SL}_2(\mathbf{C})$ and describing $P_t(\mathbf{C})$ as a conjugacy class, one deduces that $P_t^\sharp$ is Zariski-dense in $P_t(\mathbf{C})$. So $P_0^\sharp\times P_1^\sharp$ (which is a 4-dimensional real manifold) is Zariski-dense in $(P_0\times P_1)(\mathbf{C})$.

For every given nontrivial element $w$ in $\mathrm{SL}_2(\mathbf{Z})$ the set of representations vanishing on $w$ is a proper subvariety of $P_0\times P_1$, hence has dimension $\le 3$. By Zariski density of $P_0^\sharp\times P_1^\sharp$, we deduce that its intersection with $P_0^\sharp\times P_1^\sharp$ is a proper Zariski closed subset (because $\mathrm{SL}_2(\mathbf{Z})$ admits one faithful representation into $\mathrm{SL}_2(\mathbf{C})$, 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 $P_0^\sharp\times P_1^\sharp$ defining a faithful representation.