Combinatorial proofs of a theorem of Birman and Hilden and of a theorem of Magnus by the theory of automorphic sets and the respective braid group operations. (Kombinatorische Beweise eines Satzes von Birman und Hilden und eines Satzes von Magnus mit der Theorie der automorphen Mengen und der zugehörigen Zopfgruppenoperationen.) (German) Zbl 0803.20023

Summary: We deal with the well-known operation of Artin’s braid group \(B_ n\) on the free group \(F_ n\) by automorphisms, and give a proof for a theorem of Birman/Hilden (here Satz B) by showing, that the images of the generators of \(F_ n\) under \(B_ n\) are of a special form (Satz C). The theory of Brieskorn’s automorphic sets comes in here. With similar methods we give a proof of a theorem of Magnus saying that \(B_ n\) operates on a certain polynomial ring effectively by automorphisms (here Satz 9.2).


20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
Full Text: DOI


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