Topological mixing in $CAT\left(-1\right)$-spaces

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