Vol. 86, No. 1, 1980

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ISSN: 0030-8730
Groups of automorphisms of Lie groups: density properties, bounded orbits, and homogeneous spaces of finite volume

Frederick Paul Greenleaf and Martin Allen Moskowitz

Vol. 86 (1980), No. 1, 59–87
Abstract

Let G be a connected Lie group and 𝒜 an arbitrary, not necessarily connected Lie subgroup of Aut (G). For a class of groups G described below, the authors examine the bounded 𝒜-orbits in G (those with compact closure) and the group of bounded elements B(G,𝒜) = {x G : 𝒜x is bounded}. They first show that G may be split into closed, 𝒜-invariant “layers” terminating at B(G,𝒜) whose properties insure that (i) B(G,𝒜) is closed, (ii) every finite, 𝒜-invariant Borel measure must have supp μ B(G,𝒜), and (iii) if x B(G,𝒜), there is such a measure μ with x supp μ. Using these results, they prove a number of density theorems of the following sort.

Theorem 1.1. Let ℬ⊆𝒜 be arbitrary, not necessarily connected Lie subgroups of Aut (G) such that 𝒜has finite volume (or is compact). For any x G, x is bounded 𝒜x is bounded.

Theorem 1.2. Let B A be arbitrary, not necessarily connected, closed subgroups of G such that A∕B has finite volume (or is compact). Let α Aut (G) be arbitrary. Then the displacement set disp(α,B) = {α(x)x1 : x B} is bounded disp (α,A) is bounded.

The authors prove these results for G whose Levi factor is faithfully represented; there are indications that they remain true for all connected G. The proofs devolve to questions about faithful linear representations of certain nonconnected groups. Recent results of G. Hochschild show that G ×σ Aut (G) is faithfully represented if (i) the Levi factor of G is faithfully represented, and (ii) the nilradical is simply connected. In the authors’ work it is crucial to know that the representation can be chosen so that G × I (if nol all of G ×σ Aut (G)) is mapped to a closed subgroup of GL (V ); this is proved by modifying Hochschild’s proofs, coupling them with methods developed by M. Goto.

Furthermore, invariant finite Borel measures play a large role: Lie theory yields information which drastically restricts the possible locations of such measures. In crucial places the converse is true: invariant measure arguments seem necessary to obtain algebraic and geometric information about the actions.

Mathematical Subject Classification 2000
Primary: 22E15
Milestones
Received: 16 January 1979
Published: 1 January 1980
Authors
Frederick Paul Greenleaf
Martin Allen Moskowitz