Lifting wreath product extensions (Q2701619)

Let \(K\) be an Hilbertian field and let \(G\) be a finite group. Then \(G\) has the arithmetic lifting property over \(K\) if every \(G\)-extension over \(K\) is a spezialization of a geometric \(G\)-Galois branched covering of \({\mathbb{P}}_K^1\). NEWLINENEWLINENEWLINEThe main result of the paper under review states that the wreath product \(G\wr H\) has the arithmetic lifting property if \(G\) has a generic extension over \(K\) and \(H\) has the arithmetic lifting property. As a corollary we get that the semidirect product has the arithmetic lifting property if \(G\) is abelian and \(G\) and \(H\) have relatively prime orders. NEWLINENEWLINENEWLINEThese results strengthen some of the author's previous results and shorten their proofs. In contrast to the early proofs, the proofs presented here are existential and not constructive.