CAT(0) spaces on which a certain type of singularity is bounded

Abstract

In this paper, we will consider a family $\mathscr{Y}$ of complete CAT(0) spaces such that the tangent cone TC_p Y at each point p $\in$ Y of each Y $\in$ $\mathscr{Y}$ is isometric to a (finite or infinite) product of the Euclidean cones Cone(X_α) over elements X_α of some Gromov-Hausdorff precompact family {X_α} of CAT(1) spaces. Each element of such $\mathscr{Y}$ is a space presented by Gromov [4] as an example of a "CAT(0) space with "bounded" singularities". We will show that the Izeki-Nayatani invariants of spaces in such a family are uniformly bounded from above by a constant strictly less than 1.