I used this applet to figure out to
some extent when the parameters are
discrete. Here is what I found:
The group x/y/z is the (x,y,z) triangle group.
The numbers are listed so that x<=y<=z.
The element 1 is reflection in the side opposite to the angle labelled x.
The element 2 is reflection in the side opposite to the angle labelled y.
The element 3 is reflection in the side opposite to the angle labelled z
x/z/y is discrete and faithful iff neither of the
two words 3231 or 123 is elliptic.
If you believe Conjecture 1 then the first word to
go elliptic in the deformation of a triangle group
is either 123 or 3231.
There is an infinite family of discrete but not
faithful embeddings of x/y/z iff the element 3231 goes
Say that x/y/z is "good" if 3231 goes elliptic first.
Otherwise call z/y/x "bad". Some general observations:
1. x/y/z good seems to imply x/y/(z+1) good
2. x/y/y bad seems to imply x/(y+1)/(y+1) bad
3. A neighborhood of infty/infty/infty is bad.
The list of good groups: (There might be some
round-off error near the end of the 10's series.)
10/25/z z>110 (approximately)