Top degree \(\ell ^p\)-homology and conformal dimension of buildings

For a non-compact finite thickness building whose Davis apartment is an orientable pseudomanifold, we compute the supremum of the set of \(p>1\) $p>1$ such that its top dimensional reduced \(\ell ^p\) ${\ell }^p$ -cohomology is nonzero. We adapt the non-vanishing assertion of this result to any finite thickness building using the Bestvina realization. Using similar techniques, we generalize bounds obtained by Clais on the conformal dimension of some Gromov-hyperbolic buildings to any such building.