Group actions by automorphisms of compact groups

Abstract

Throughout this book the term compact group will denote a compact, metrizable group, and a compact Lie group will be a compact (possibly finite) subgroup of some finite-dimensional matrix group over the complex numbers. Any metric δ on a compact group X is assumed to be invariant, i.e. δ(x, x’) = δ(yx, yx’) = δ(xy, x’y) for all x, x’, y ∈ X, and to induce the topology of X. The identity element of a group X will usually be written as 1, 0, 1 _X , or 0 _X , depending on whether X is multiplicative or additive, and whether there is danger of confusion. If X is a topological group we write X° for its connected component of the identity, C(X) for its centre, Aut(X) for the group of continuous automorphisms of X, and Inn(X) ⊂ Aut(X) for the normal subgroup of inner automorphisms of X. The trivial automorphism of X is denoted by id _X =1 _Aut(X), and we set Out(X) = Aut(X)/Inn(X). If X is compact and δ is a metric on X we define a metric δ on Aut(X) by setting

$$ \partial \left( {\alpha, \beta } \right)\mathop{{\max }}\limits_{{x \in X}} \left( {\partial \left( {\alpha (x),\beta (x)} \right) + } \right)\partial \left( {{\alpha^{{ - 1}}}(x),{\beta^{{ - 1}}}\left( {x)} \right)} \right) $$

for all α, β ∈ Aut(X); the topology on Aut(X) induced by δ is called the uniform topology. If X is a compact Lie group then Aut(X) is again a Lie group in the uniform topology, Inn(X) is an open subgroup of X, and Out(X) is therefore discrete. For an arbitrary compact group X, the group Out(X) is zero-dimensional in the uniform topology ([32]).