Patching on Berkovich spaces. (Q969276)

The author presents several results on the inverse Galois problem. He uses geometric patching in Berkovich spaces to prove that if \(K\) is a complete ultrametric field with non-trivial valuation, then every finite group is a Galois group over the field \(K(T)\). He gives a new, purely geometric proof of the theorem of Harbater that every finite group is a Galois group over a field of convergent arithmetic power series.