In each of the two groups, the common value of x2 = y3 = z7 = 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 | x2 = y3 = z7 = 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.
 H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Springer-Verlag, 1957, 1965, 1972, 1980.
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