Locally compact groups with many automorphisms
by
Markus Stroppel
Mathematisches Institut B, Universität Stuttgart, 70550 Stuttgart, Germany

We study locally compact groups under the additional assumption that the number of orbits under the group of all topological automorphisms is bounded by a suitable cardinal number b. Such assumptions turn out to be restrictive if the group is compact or connected and b is countable, or if b <= 3, or if the group is abelian and b is finite. We indicate some typical results:

Explicit descriptions for G can be given if b=3: typical examples are certain generalized Heisenberg groups, or semidirect extensions of vector groups by suitable automorphisms. For a compact or connected locally compact group G, even b < 2\sp\aleph₀ turns out to be very restrictive. If G is compact then b < 2\sp\aleph₀ yields that G is totally disconnected, and b < \aleph₀ yields that G is a torsion group of bounded exponent, and possesses an open solvable characteristic subgroup (with finite quotient, of course). If G is locally compact connected then b < 2\sp\aleph₀ implies that G is a nilpotent, simply connected Lie group

Date received: May 4, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-18.