Fibred coarse embedding into non-positively curved manifolds and higher index problem

Let X=⨆n=1∞Xn be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. In this paper, we show that the coarse Novikov conjecture holds for X, if X admits a fibred coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature. This includes a large class of expander graphs with geometric property (T).