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

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.

arXiv admin note: text overlap with arXiv:2307.05640