|Date||Sep 20, 2022|
Let x be a complex number. Let A be an upper triangular matrix in SL(2,C) whose non-zero entries are 1, and B(x) a strict lower triangular matrix in SL(2,C) whose (2,1)-entry is x. Then, x is called a relation number if the group generated by A and B(x) is not a rank 2 free group. The characterization problem of the relation numbers is a long-standing conjecture in the group theory. This problem has been studied by many mathematicians including Rimhak Ree. In this talk, I introduce a topological and dynamical tool, the generalized Farey graphs. This gives a new equivalent definition for relation numbers. Then, I show how the generalized Farey graph can be applied to find a sequence of relation numbers which are unknown before.