In each of the two groups, the common value
of *x*^{2} = *y*^{3}
= *z*^{7} = *xyz* generates an
infinite cyclic central subgroup; so it suffices to show
that the homomorphism between the factor-groups by this
subgroup is an isomorphism; i.e., between
⟨*x, y, z* | *x*^{2} = *y*^{3}
= *z*^{7} = *xyz* = 1⟩ and its
image in PSL(2, **R**).
The former group, I have learned, is denoted *D*(2,3,7).
The groups *D*(*l,m,n*) with general
exponents *l, m, n,* in place of 2, 3, 7 are called
von Dyck groups (or simply Dyck groups);
and the von Dyck groups *D*(2,*m,n*) have natural
descriptions as symmetry groups of tesselations.
In particular, *D*(2,3,7) is the group of proper
symmetries of the tesselation of the hyperbolic plane by
7-gons meeting 3 at a vertex [12, section 5.3].
I am told that isometries of the hyperbolic plane can be
represented by 2×2 matrices, and that this leads
to the desired isomorphism between *D*(2,3,7) and
the desired subgroup of PSL(2, **R**), but I do not
at this time know references for the relevant facts.

[12] H. S. M. Coxeter and W. O. J. Moser,
*Generators and relations for discrete groups*, Springer-Verlag,
1957, 1965, 1972, 1980.