Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are there examples showing that this inequality is not a sufficient condition? Are there methods to prove that two given elements do NOT generate a discrete subgroup?


Background: A recently published paper <http://xm10402.reportworld.co.kr/data/paper/view/2882/P2881838.html> claims (in Theorem 4.5. in the appendix) that a group generated by two hyperbolic matrices A,B of positive trace is discrete if and only if their axes intersect and the trace of the commutator [A,B] is smaller than -2. This theorem seems a bit strong to me but it is not obvious to me how to construct counterexamples. In fact the condition on the trace of the commutator automatically guarantees that Jorgensen's inequality holds, so the first check for non-discreteness must fail.



Matrices generating non-discrete subgroups of SL(2,R)
For more than you ever dreamed possible on the subject, see Jane Gilman's 


MR1290281 (97a:20082) 
Gilman, Jane(1-RTG2)
Two-generator discrete subgroups of PSL(2,R). (English summary) 
Mem. Amer. Math. Soc. 117 (1995), no. 561, x+204 pp. 
20H10 (22E40 30F35) 


For a more concise paper which will answer your question, see:


Rosenberger, Gerhard(D-DORT)
All generating pairs of all two-generator Fuchsian groups. 
Arch. Math. (Basel) 46 (1986), no. 3, 198–204. 
20H10 


