The automorphism groups of compact homogeneous spaces
Let \(G\) be a connected Lie group and let \(H\) be a closed subgroup of \(G\). The author studies the group \(A := \mathrm{Aut}_G (G/H) \cong N_G(H) / H \) of \(G\)-equivariant automorphisms of the homogeneous space \(M = G/H\), mainly for the case that \(M\) is compact. The paper deals with the following topics: General properties; extreme cases, when \( \dim A \) or \(\dim M - \dim A \) is small; methods to compute \(A\) with the goal to reduce the computation to Lie algebra computations and computations of centralizers of lattices. The author exemplifies this for the case that \(\pi_1(M)\) is solvable, based on his earlier work [\textit{V. V. Gorbatsevich}, in: Geom. Metody Zadachakh Algebry Anal. 1981, 71--87 (1981; Zbl 0499.57015); Vopr. Teor. Grupp Gomologicheskoj Algebry 1982, 13--28 (1982; Zbl 0583.57014); ibid. 1985, 93--103 (1985; Zbl 0735.57023); in: A. L. Onishchik (ed.), Questions of group theory and homological algebra. Collection of scientific works. Yaroslavl': Yaroslavskij Gosudarstvennyj Universitet. 88--98 (1991; Zbl 0826.57015)]. In the last chapter he discusses centralizers and normalizers of lattices. Unfortunately, Theorem~7 is false. In this theorem the author claims that for a semisimple Lie group \(G\) without compact quotients and a cocompact lattice \(\Gamma\) in \(G\) the automorphism group \(A := \mathrm{Aut}_G(G/H) \cong N_G(H)/H\) is finite and abelian. The finiteness claim is correct. The claim about the commutativity would imply that for every cocompact lattice \(\Delta\) in \(G\) every finite homomorphic image \(\Delta/\Gamma \subset A\) is abelian, which is not true, e.g., for fundamental groups \(\Delta \subset \mathrm{PSL}(2,\mathbb{R})\) of compact Riemann surfaces.
scientific article; zbMATH DE number 6658517