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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).

MSC:

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

[1] Artin, E., Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, 4, 47-70 (1925) · JFM 51.0450.01 · doi:10.1007/BF02950718
[2] Artin, E., Theory of Braids, Ann. of Math., 48, 2, 101-126 (1947) · Zbl 0030.17703 · doi:10.2307/1969218
[3] Birman, J. S., Braids, Links and Mapping Class Groups, Ann. of Math. Stud. (1974), Princeton N.J.: Princeton Univ. Press, Princeton N.J.
[4] Birman, J. S.; Hilden, H. M., On Isotopies of Homeomorphisms of Riemann Surfaces, Ann. of Math., 97, 2, 424-439 (1973) · Zbl 0237.57001 · doi:10.2307/1970830
[5] Brieskorn, E., Automorphic Sets and Braids and Singularities, Braids, Proc. of a Summer Conf. (Univ. Cal.), Santa Cruz 1986, Contemporary Math.(AMS) (1988), Herausgeber: J.S. Birman, Herausgeber · Zbl 0716.20017
[6] Chow, W. L., On the Algebraic Braid Group, Ann. of Math., 49, 2, 654-658 (1948) · Zbl 0033.01002 · doi:10.2307/1969050
[7] Hurwitz, A., Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., 39, 1-61 (1891) · JFM 23.0429.01 · doi:10.1007/BF01199469
[8] Johnson, D. L., On a Problem of Magnus, J. of Algebra, 79, 121-126 (1982) · Zbl 0495.20017 · doi:10.1016/0021-8693(82)90320-9
[9] Joyce, D., A Classifying Invariant of Knots, the Knot Quandle, J. Pure Appl. Algebra, 23, 37-65 (1982) · Zbl 0474.57003 · doi:10.1016/0022-4049(82)90077-9
[10] B. Krüger, Automorphe Mengen und die Artinschen Zopfgruppen (Dissertation 1989), Bonner Mathematische Schriften Nr.207. · Zbl 0745.20036
[11] Loos, O., Spiegelungsräume und homogene symmetrische Räume, Math. Z., 99, 141-170 (1967) · Zbl 0148.17403 · doi:10.1007/BF01123745
[12] Magnus, W., Rings of Fricke Characters and Automorphism Groups of Free Groups, Math. Z., 170, 91-103 (1980) · Zbl 0433.20033 · doi:10.1007/BF01214715
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