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

Abstract We prove that if a sequence of geodesically complete CAT(0)-spaces $$X_j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $$X_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> , then the dimension of the maximal Euclidean factor splitted off by $$X_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> and $$X_j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> is the same, for j big enough. In other words, no additional Euclidean factors can appear in the limit.