Convergence of isometries, with semicontinuity of symmetry of Alexandrov spaces

TL;DR

This paper investigates the convergence behavior of isometry groups in Alexandrov spaces, establishing conditions under which symmetry groups converge and can only increase, with implications for geometric analysis.

Contribution

It proves that under certain bounds, the convergence of isometry groups in Alexandrov spaces is by Lie homomorphisms, and symmetries increase with curvature and volume bounds.

Findings

Convergence by Lie homomorphisms under dimension bounds.

Symmetries can only increase with curvature and volume bounds.

Establishment of semicontinuity of symmetry in Alexandrov spaces.

Abstract

The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this result to convergence by monomorphisms, so that symmetries can only increase on passing to the limit.