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The holomorphic automorphism group of the complex disk. (English) Zbl 0799.20032
Each holomorphic permutation of the open disk in \(\mathbb{C}\) with positive radius \(c\) and centre 0 can be written as the product of a rotation and a so-called gyration \(z\mapsto(a+z)/(1+\overline{a}z/c^ 2)\). Properties of this product decomposition are axiomatized, giving rise to the notions of a gyrogroup and a gyrosemidirect product.

20E22 Extensions, wreath products, and other compositions of groups
22E43 Structure and representation of the Lorentz group
20N05 Loops, quasigroups
30H05 Spaces of bounded analytic functions of one complex variable
32A38 Algebras of holomorphic functions of several complex variables
83A05 Special relativity
30A05 Monogenic and polygenic functions of one complex variable
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