On a theorem of Auslander (Q2353478)

A result by L. Auslander at the end of the 60s of the last century sets that under the following conditions: \(G\), a Lie group; \(R\), a closed connected solvable normal subgroup of \(G\); \(\pi : G \mapsto G/R\), the natural map; \(H\), a closed subgroup of \(G\) such that \(H^0\), the identity component of \(H\), is solvable and \(U = \overline{\pi(H)}\) the closure of \(\pi(H)\), then the identity component \(U^0\) of \(U\) is solvable. This result was later extended by \textit{S. P. Wang} [Am. J. Math. 92, 708--724 (1970; Zbl 0223.22008)] to the case that \(R\) is not necessarily connected. In the paper under review, the author gives a short proof of a new extension of the result by Auslander by replacing \(\pi\) by any continuous homomorphism \(f\) with solvable kernel. In particular, the kernel of \(f\) needs not be connected and the quotient \(G / \mathrm{ker}(f)\) needs not have the quotient topology and may be dense in some larger group. Then, the author extends the original result further to the case that \(G\) has a virtually solvable group of connected components.