Topological mixing in 𝐶𝐴𝑇(-1)-spaces

If X X is a proper C A T ( − 1 ) CAT\left ( -1\right ) -space and Γ \Gamma a non-elementary discrete group of isometries acting properly discontinuously on X , X, it is shown that the geodesic flow on the quotient space Y = X / Γ Y=X/\Gamma is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂ X \partial X and the non-wandering set of the flow equals the whole quotient space of geodesics G Y := G X / Γ GY:=GX/\,\Gamma (the latter being redundant when Y Y is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete C A T ( − 1 ) CAT\left ( -1\right ) -spaces by a one-ended group of isometries and (C) finite n n -dimensional ideal polyhedra.