Locally compact groups with many automorphisms (Q2769826)

In [J. Algebra 199, 528-543 (1998; Zbl 0889.22005)] the author determined all connected, locally compact groups \(G\) such that the number \(\omega\) of orbits of \(\Aut G\) equals 2. Together with \textit{H. Mäurer} [Geom. Dedicata 68, 229-243 (1997; Zbl 0890.20025)] he considered discrete groups with \(\omega= 3\). Among other things, they showed that a nilpotent group of prime exponent with \(\omega= 3\) is a (generalized) Heisenberg group, i.e., a direct product of two vector groups over the same field \(F\), with multiplication defined by \((v,y)\circ (w,z)= (v+ w,y+\beta(v, w)+ z)\), where \(\beta\) is a symplectic bilinear map. Not all Heisenberg groups have \(\omega= 3\), but those with \(F=\mathbb{R}\) and \(\omega= 3\) have been classified by the author [Forum Math. 11, 659-672 (1999; Zbl 0928.22008)].NEWLINENEWLINENEWLINEIn the present paper, bounds on \(\omega\) and their effect on the structure of a topological group are studied systematically. For a locally compact, not totally disconnected group with \(\omega= 3\), only the following possibilities exist: \(G\) is (1) a connected Heisenberg group or (2) a product of \(\mathbb{R}^n\) and some discrete vector group over \(\mathbb{Q}\), or (3) the extension of \(\mathbb{R}^n\) by inversion or of \(\mathbb{C}^n\) by a scalar map of order 3, or (4) the commutator group and the center of \(G\) are both equal to \(G_1\) (the connected component), \(G/G_1\) is a discrete rational vector group and \(G\) is nilpotent, or (5) \(G\) equals its commutator group, \(G_1\) is the center, and \(G/G_1\) is torsion free; examples of type (5) are not known.NEWLINENEWLINENEWLINEUsing Pontryagin duality, it is shown that compact, connected groups have \(\omega\geq 2^{\aleph_0}\). Using the solution of Hilbert's fifth problem, it is then proved that a locally compact connected group with \(\omega< 2^{\aleph_0}\) is a nilpotent, simply connected Lie group. The compact totally disconnected groups with \(\omega= 3\) are described (they are all solvable), and a compact group with \(\omega= 4\) is either solvable or the alternating group \(A_5\).NEWLINENEWLINENEWLINEThere are further results on Abelian topological groups such that \(\omega\) is finite. For example, in the locally compact case, \(G\) is a direct product of a group of type (2) and \(\text{Comp}(G)\), the union of all compact subgroups; \(G_1\) admits a closed direct complement, which is described in more detail.