Bergman's amalgamation problem over the circle

We prove that $S^1$, $\mathrm{S}\mathrm{U}(2)$, and $\mathrm{S}\mathrm{O}(3)$ are compact amalgamation bases: for each of these groups, any two compact groups with specified embedded copies of it embed in a compact group in which those copies agree. For compact Lie groups, the amalgam can be chosen to be a finite-dimensional unitary group. This research was conducted with substantial AI assistance, including in the development of mathematical constructions and proof arguments.