Appendix: The Dehn-Nielsen Theorem

John Stillwell

A chapter in Papers on Group Theory and Topology, 1987, pp 363-396 from Springer

Abstract: Abstract One important theorem attributed to Dehn does not appear in his published work, and his proof of it seems to be lost. This is the theorem that every automorphism of the fundamental group πl (S) of a closed orientable surface S can be induced by a homeomorphism of S. The earliest known proof appears in Nielsen [1927], and consists of two parts. The first, which is in §9 of Nielsen’s paper and expounded below as Theorem 5, is quite elegant and is attributed to Dehn. The second, in §23, is agonisingly long and is apparently Nielsen’s own, though he still credited the theorem to Dehn. As was mentioned in the introduction to Paper 7, Nielsen in 1931 found a replacement for §23 which was much simpler, but he did not publish it. We give a similar proof below as Theorem 6.

Keywords: Fundamental Group; Homotopy Class; Hyperbolic Plane; Closed Geodesic; Hyperbolic Geometry (search for similar items in EconPapers)
Date: 1987
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DOI: 10.1007/978-1-4612-4668-8_17

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