Higher order Dehn functions for horospheres in products of Hadamard spaces

TL;DR

This paper establishes sharp bounds on higher order Dehn functions for horospheres in products of Hadamard spaces and for certain lattices, revealing their geometric and algebraic properties in high dimensions.

Contribution

It provides new bounds on Dehn functions for horospheres in products of Hadamard spaces and for lattices acting on these spaces, extending previous results to higher dimensions.

Findings

Horospheres are Lipschitz-(r-2)-connected in product spaces.

Sharp bounds on higher order Dehn functions are established.

Irreducible rank-one lattices are undistorted up to dimension r-1.

Abstract

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 $k\le r-2$.