Isometry groups of three-dimensional Lie groups

TL;DR

This paper computes the isometry groups of left-invariant metrics on three-dimensional non-unimodular Lie groups, analyzes their symmetry properties, and explores the structure of the moduli space of such metrics.

Contribution

It provides a complete characterization of isometry groups for these Lie groups and links symmetry indices to the structure of the metric moduli space.

Findings

Full isometry groups are determined for all such Lie groups.

The index of symmetry of these metrics is explicitly computed.

Singularities in the moduli space are contained in metrics with maximal symmetry.

Abstract

We compute the full isometry group of any left invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism, is contained in the subspace of classes of metrics with maximal index of symmetry.