The Theory and Practice of Conformal Geometry by Krantz Steven G.;

Author:Krantz, Steven G.; [Isteven G. Krantz]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2015-02-26T16:00:00+00:00

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1These issues are related to Hilbert’s fifth problem, which asked whether any locally Euclidean group is a Lie group. See [BRO] and [TAU].

2It should be stressed that it is a priori clear that these two domains cannot be conformally equivalent because they are not even topologically equivalent. After all, D is simply connected while A is not. But our point is to see how automorphism groups can be used to obtain useful results.

3It is also worth noting that one automorphism group (for the annulus) is abelian while the other (for the disc) is not. Details are left for an exercise.

4It is worth noting that the conformal self-maps of the plane are just those conformal self-mappings of the sphere that map ∞ to ∞. When we pass from mappings of the plane to mappings of the sphere, we have the extra latitude of moving the point at ∞ to another point on the sphere—and that gives two more degrees of freedom. That is why dim(Aut ) = 6 and dim(Aut ()) = 4.

5For example, in the simple case that the domain is the annulus A = {z ∈ : 1/2 < |z| < 2}, the automorphism group is the set of all rotations plus the inversion z 1/z plus compositions of the two. Topologically, this automorphism group is two circles. So the automorphism group has two connected components.

6Here the orbit of P is defined to be {ϕ(P) : ϕ ∈ Aut (U)}.

7A group is said to be nilpotent if there is an integer K so that any commutator of order K in the group is the identity.

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