  The Lodha-Moore group $G$ first arose as a finitely presented counterexample
to von Neumann's conjecture. The group $G$ acts on the unit interval via
piecewise projective homemorphisms. A result of Lodha shows that $G$ in fact
has type $F_{\infty}$.
  Here we will describe $G$ as a group that is "locally determined" by an
inverse semigroup $S_{2}$, in the sense of the author's joint work with Hughes.
The semigroup $S_{2}$ is generated by three linear fractional transformations
$A$, $B$, and $C_{2}$, where $A$ and $B$ are elliptical transformations of the
hyperbolic plane and $C_{2}$ is a hyperbolic translation. Following a general
procedure delineated by Farley and Hughes, we offer a new proof that $G$ has
type $F_{\infty}$. Our proof simultaneously shows that various groups acting on
the line, the circle, and the Cantor set have type $F_{\infty}$. We also prove
analogous results for the groups that are locally determined by an inverse
semigroup $S_{3}$, which shares the generators $A$ and $B$ with $S_{2}$, but
replaces $C_{2}$ with a different hyperbolic translation $C_{3}$.
