GH-convergence of CAT$(0)$-spaces: stability of the Euclidean factor

TL;DR

This paper proves that in the Gromov-Hausdorff convergence of geodesically complete CAT(0)-spaces with cocompact isometry groups, the Euclidean factor's dimension remains stable, preventing new Euclidean factors from emerging in the limit.

Contribution

It establishes the stability of the Euclidean factor's dimension under GH-convergence for a class of CAT(0)-spaces with cocompact isometry groups.

Findings

The Euclidean factor dimension is preserved in the limit.

No new Euclidean factors appear in the Gromov-Hausdorff limit.

Stability result applies to geodesically complete CAT(0)-spaces with cocompact isometry groups.

Abstract

We prove that if a sequence of geodesically complete CAT$(0)$-spaces $X_j$ with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $X_\infty$, then the dimension of the maximal Euclidean factor splitted off by $X_\infty$ and $X_j$ is the same, for $j$ big enough. In other words, no additional Euclidean factors can appear in the limit.