Isometry groups of Alexandrov spaces

TL;DR

This paper investigates the symmetry groups of Alexandrov spaces, establishing maximal isometry group dimensions, characterizing when such spaces are Riemannian manifolds, and identifying gaps in possible symmetry dimensions.

Contribution

It determines the maximal dimension of isometry groups for Alexandrov spaces and characterizes when these spaces are Riemannian manifolds based on symmetry.

Findings

Maximal isometry group dimension for Alexandrov spaces is established.

Spaces with maximal symmetry are shown to be Riemannian manifolds.

A gap in possible isometry group dimensions is identified.

Abstract

For an Alexandrov space (with curvature bounded below), we determine the maximal dimension of its isometry group and show that the space is isometric to a Riemannian manifold, provided the dimension of its isometry group is maximal. We also determine a gap in the possible dimensions of the isometry groups and show that if the Alexandrov space is symmetric, then it is isometric to a Riemannian manifold.