Maximal Symmetry and Unimodular Solvmanifolds

TL;DR

This paper investigates the symmetry properties of unimodular solvable Lie groups, demonstrating the existence of metrics with maximal symmetry and linking Ricci solitons to maximal isometry groups.

Contribution

It proves that unimodular solvable Lie groups always admit a metric with maximal symmetry and connects Ricci solitons to the maximal isometry group in such cases.

Findings

Unimodular solvable Lie groups have metrics with maximal symmetry.

Ricci solitons on these groups have maximal isometry groups.

Maximal symmetry is achieved even in the unimodular case, unlike the non-unimodular case.

Abstract

Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and prove that there is always some metric with maximal symmetry. Further, if the group at hand admits a Ricci soliton, then it is the isometry group of the Ricci soliton which is maximal.