Infinitesimal maximal symmetry and Ricci soliton solvmanifolds (Q6571614)

The authors consider three notions of maximal symmetry for the class of left-invariant Riemannian metrics on a given connected Lie group: maximal, almost maximal, and infinitesimally maximal symmetry. These notions essentially measure the extent to which the isometry group (resp.~its identity component, or its Lie algebra) of one such metric contains that of all the other metrics, up to a suitable conjugation. This article is a follow-up on previous work by the same authors [\textit{C. S. Gordon} and \textit{M. R. Jablonski}, J. Differ. Geom. 111, No. 1, 1--38 (2019; Zbl 1415.53037)], in which they showed that Einstein solvmanifolds have maximal symmetry (Theorem 1.2). The main result of the present article is that for Ricci soliton metrics, infinitesimal maximal symmetry holds (Theorem 1.7). The authors also show that a given solvable group can act simply-transitively by isometries on at most one Ricci soliton up to homothety (Corollary 1.6).\N\NA key problem in the study of Riemannian solvmanifolds is the lack of uniqueness for the choice of a transitive solvable group of isometries. This group is only `canonically defined' when it is nilpotent, or more generally unimodular and completely solvable. In general, it is hard to dermine geometric properties in terms of Lie algebraic conditions in other cases. The authors circumvent this difficulty by establishing an equivalence relation on the class of simply-connected solvable Lie groups, which can be defined algebraically, but also equivalently in geometric terms (Theorem 1.4, (ii)). This allows them to reduce the proof of the main result to the case of completely solvable groups. For the latter they use an argument that uses the Einstein one-dimensional extension construction [\textit{J. Lauret}, J. Reine Angew. Math. 650, 1--21 (2011; Zbl 1210.53051)], and essentially reduces Theorem 1.7 to Theorem 1.2.\N\NThe authors leave an interesting open question: Does every Lie group admit an infinitesimally maximally symmetric left-invariant metric? It also remains an interesting question to determine whether the completely solvable case in the proof of the main theorem can be established by Ricci flow (cf. [\textit{C. Böhm} and \textit{R. A. Lafuente}, Adv. Math. 352, 516--540 (2019; Zbl 1423.53079)]).