I'm looking into matrix representations of Hurwitz groups, beginning with 2x2 matrices. There are many representations with finite characteristic, namely the ${\rm PSL}(2,p^n)$ groups, where $n=1$ if $p=7$ or $p=1(\operatorname{mod} 7)$, and $n=3$ otherwise. However, what about in characteristic zero?


If we define matrices $A=\begin{pmatrix}0&-1 \\ 1&0 \\\end{pmatrix}, B=\begin{pmatrix}0&-r^{-1} \\ r&1 \\\end{pmatrix}$, where $r^7=1$, then one can verify that $A^2=B^3=(AB)^7=I$ (working in the projective special linear group). The group $H=\langle A,B\rangle$ clearly has all the finite characteristic 2x2 matrix representations as quotients, but I doubt it has the other finite Hurwitz groups as quotients. Therefore, it would seem that $H$ must have some hidden extra relations.


My question is: What are those hidden relations? Or am I mistaken and the relations: $A^2=B^3=(AB)^7=I$ are all there is?



Relators of the "most general" 2x2 matrix Hurwitz group
I think the answer to your question, is yes, this is a faithful projective representation of the triangle group $G = \langle a,b \mid a^2,b^3,(ab)^7 \rangle$, although I have only a hazy understanding of the details.


In Corollary 3.2 of [this paper by Plesken](https://www.sciencedirect.com/science/article/pii/S0021869306001189) it is proved that, up to Galois conjugacy, there is a unique complex irreducible $2$-dimensional projective representation of the group, and he points out before the proof that there can be no faithful linear $2$-dimensional representation, because $G$ is perfect and $-I\_2$ is the only element of order $2$ with determinant $1$. The representation that he writes down gives looks similar to yours.


Then, near the beginning of the proof of Theorem 4.2 of the paper, Plesken remarks that Hurwitz studied a projective representation of $G$ into ${\rm PSL}\_2({\mathbb R})$, which is faithful, so that appears to imply that your representation is faithful.


