Isometry groups with radical, and aspherical Riemannian manifolds with large symmetry I

TL;DR

This paper explores the structure of aspherical Riemannian manifolds with large symmetry, introducing the infrasolv tower concept and constructing examples with high symmetry that lack locally homogeneous metrics.

Contribution

It defines the infrasolv tower for aspherical manifolds and constructs examples with large symmetry that do not support locally homogeneous metrics.

Findings

Introduces the infrasolv tower as a measure of symmetry.

Constructs examples of manifolds with large symmetry but no locally homogeneous metrics.

Shows the relationship between the infrasolv tower and the manifold's symmetry.

Abstract

Every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infrasolv fibers which is called its infrasolv tower. The tower arises from the solvable radicals of isometry group actions on the universal covers. Its length and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. We say that the manifold has large symmetry if it admits an infrasolv tower whose base is a locally homogeneous space. We construct examples of aspherical manifolds with large symmetry, which do not support any locally homogeneous Riemannian metrics.