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.
Also, in the last sentence of the paper before the Acknowledgement, I ask whether there exist nontrivial finitely generated right orderable groups having both finite abelianization and trivial center. Yves Cornulier and Dave Witte Morris have pointed out to me that such groups exist: The free product of two nontrivial right orderable perfect groups will have this property, with abelianization not merely finite but trivial. The group constructed in  has the still stronger property of being simple.
 H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Springer-Verlag, 1957, 1965, 1972, 1980.
 James Hyde and Yash Lodha, Finitely generated infinite simple groups of homeomorphisms of the real line, 2018, https://arxiv.org/abs/1807.06478 .
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