Some thoughts and experiments on Bergman's compact amalgamation problem

We study the question whether copies of $S^1$ in $\mathrm{S}\mathrm{U}(3)$ can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. Considerable computational experiments suggest that the answer is positive in this case. We obtain a positive answer for a relaxed problem using theoretical considerations.