Siberian Mathematical Journal, Vol. 42, No. 6, pp. 1036–1046, 2001 Original Russian Text Copyright c  2001 Gorbatsevich V. V. TRANSITIVE ISOMETRY GROUPS OF ASPHERIC RIEMANNIAN MANIFOLDS V. V. Gorbatsevich UDC 519.46 In this article we study the isometry groups Iso(M ) of aspheric Riemannian manifolds M . A manifold M is aspheric if all its homotopy groups π i (M ) vanish for i ≥ 2. This amounts to the fact that the universal covering  M of M is contractible to a singleton. Here we consider homogeneous aspheric Riemannian manifolds, i.e., aspheric manifolds endowed with a Riemannian metric f and a transitive action of some Lie group G of isometries (a subgroup of the group Iso(M,f ) of all isometries of the manifold; the latter group is well known to be a Lie group). Originally the author was only interested in Riemannian solvmanifolds, i.e., homogeneous Rieman- nian spaces of solvable Lie groups G. Riemannian solvmanifolds are the subject of investigations in many articles, but only isolated examples or narrow classes of such manifolds were considered in those articles or their authors study all possible transitive groups of isometries on M (see [1–4]). The description of [2] for all connected subgroups of Iso(M ) transitive on a given solvmanifold M is rather intricate and the proof in [2] is technically difficult. In the case under consideration the authors of [2] factually in- troduced the notion of splitting of a Lie group which was, however, well known already at that time (see more detail in the survey [5]). The author of the present article was interested in the very group Iso(M ); providing description for its structure (by using a certain reduction) turned out to be much easier than describing the structure of all its transitive subgroups. As it seems to the author, the preceding articles about Riemannian solvmanifolds did not reveal the essence of the structure of the groups Iso(M ) for Riemannian solvmanifolds. In this article we repeatedly use the notion of a maximally connected triangular subgroup. We manage to reveal the cause for the special role that the triangular Lie groups play in the theory of Riemannian solvmanifolds, the theory of Riemannian Lie algebras (i.e., Lie algebras endowed with an inner product), and the theory of Thurston geometries (where solvable geometries turn out to be in some sense triangular). So far triangular Lie groups only played a secondary role, and it now appears that they perform a leading role. Observe that at present the theory of Thurston geometries is not complete even in the three-dimensional case, since it is only homogeneous spaces themselves that are classified but not the metrics on them (among which there exist different, i.e., nonisometric metrics). Our approach makes it possible to distinguish “special” metrics for Thurston geometries and opens up an opportunity to describe the corresponding geometries in more detail (but these questions are not discussed in the present article). While studying Riemannian solvmanifolds, we have discovered that some results are valid (sometimes after a suitable rewarding) in a wider situation: for arbitrary aspheric homogeneous Riemannian spaces. Every solvmanifold is aspheric (see below), but the class of aspheric homogeneous Riemannian solvmani- folds is distinguished in the class of all homogeneous Riemannian spaces by purely topological conditions (π i (M ) = 0 for i ≥ 2), whereas the recognition of Riemannian solvmanifolds is not purely topological (it relies on the requirement of existence of a transitive solvable group of isometries). Moreover, it turns out that the class of Riemannian solvmanifolds cannot be distinguished by purely topological conditions in principle, because it follows from Theorem 1 to be proved below that topologically the classes of aspheric homogeneous Riemannian spaces and Riemannian solvmanifolds are indistinguishable (although these two classes of course differ geometrically). Proofs of the main results in the aspheric case happened to be slightly more laborious than those of The research was supported by the Russian Foundation for Basic Research (Grant 98–01–00329). Moscow. Translated from Sibirski˘ ı Matematicheski˘ ı Zhurnal, Vol. 42, No. 6, pp. 1244–1258, November–December, 2001. Original article submitted May 5, 2000. 1036 0037-4466/01/4206–1036 $25.00 c  2001 Plenum Publishing Corporation