Higher order Dehn functions for horospheres in products of Hadamard spaces

Let $X$ be a product of $r$ locally compact Hadamard spaces. In this note we prove that the horospheres in $X$ centered at regular boundary points of $X$ are Lipschitz-$(r-2)$-connected. Using the filling construction by R.~Young in \cite{MR3268779} this gives sharp bounds on higher order Dehn functions for such horospheres. Moreover, if $\Gamma\subset\Is(X)$ is a lattice acting cocompactly on $X$ minus a union of disjoint horoballs, we get a sharp bound on higher order Dehn functions for $\Gamma$. We therefore deduce that apart from the Hilbert modular groups already considered by R.~Young every irreducible $\QQ$-rank one lattice acting on a product of $r$ symmetric spaces of the noncompact type is undistorted up to dimension $r-1$ and has $k$-th order Dehn function asymptotic to $V^{(k+1)/k}$ for all $k\le r-2$.