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

CONJECTURE 1: 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.

CONJECTURE 2: There is an infinite family of discrete but not faithful embeddings of x/y/z iff the element 3231 goes elliptic first.

*********************

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.)

x/y/z x<10

10/y/z y<15

10/15/z z>15

10/16/z z>16

10/17/z z>18

10/18/z z>20

10/19/z z>23

10/20/z z>26

10/21/z z>30

10/22/z z>35

10/23/z z>43

10/24/z z>58

10/25/z z>110 (approximately)

11/11/z z>11

11/12/z z>13

11/13/z z>15

11/14/z z>18

11/15/z z>22

11/16/z z>20

11/17/z z>48

12/12/z z>16

12/13/z z>21

12/14/z z>32

13/13/z z>39