From Möbius to Gyrogroups
From Möbius to Gyrogroups
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DOI：
10.1080/00029890.2008.11920506
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发表时间：
2008-02
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影响因子：
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通讯作者：
A. Ungar
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文献类型：
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作者：
A. Ungar
Abraham A. UngarAmer. Math. Monthly, Vol. 115(2), (2008), pp. 138–144.1. IntroductionThe evolution from Mo¨bius to gyrogroups began in 1988 [12], and is still ongoingin [14, 15]. Gyrogroups, a natural generalization of groups, lay a fruitful bridgebetween nonassociativealgebraand hyperbolicgeometry, just asgroupslayafruitfulbridge between associative algebra and Euclidean geometry. More than 150 yearshave passed since the German mathematician August Ferdinand Mo¨bius (1790–1868) first studied the transformations that now bear his name [9, p. 71]. Yet, therich structure he thereby exposed is still far from being exhausted, as the evolutionfrom Mo¨bius to gyrogroups demonstrates.2. FromM¨obiusMo¨bius transformations of the complex open unit disc D= {z ∈ C: |z| < 1} ofthe complex plane Care studied in most books on function theory of one complexvariable. According to [10, p. 176], these are important for at least two reasons:(i) they play a central role in non-Euclidean geometry [15], and (ii) they are theonly automorphisms of the disc. Ahlfors’ book [1],
