Abstract:
Determining which pairs of parabolic elements of SL(2, C) generate a free group is a long-standing problem.
We will explore and prove certain results before focusing on a collection of subgroups of particular interest, which is conjectured to consist of only non-free subgroups.
While many of those subgroups are known to be non-free, all methods to find such subgroups rely on the presence of relators of short syllable length. We will explain how to construct subgroups with no such short relators, implying that existing methods might not be suitable for solving the conjecture.
