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.

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 [13] has the still stronger property of being simple.

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

[13] James Hyde and Yash Lodha,
*Finitely generated infinite simple groups of homeomorphisms
of the real line,* 2018,
https://arxiv.org/abs/1807.06478 .