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

TL;DR

This paper investigates the top degree $\, ext{l}^p$-homology of buildings, establishing conditions for non-vanishing cohomology and deriving bounds on conformal dimension for Gromov-hyperbolic buildings using geometric and algebraic techniques.

Contribution

It computes the supremum of p for which top $\, ext{l}^p$-cohomology is nonzero in certain buildings and generalizes bounds on conformal dimension to a broader class of buildings.

Findings

Identifies the supremum of p with non-vanishing top $\, ext{l}^p$-cohomology.

Extends non-vanishing results to all finite thickness buildings via Bestvina realization.

Provides generalized bounds on conformal dimension for Gromov-hyperbolic buildings.

Abstract

For a non-compact finite thickness building whose Davis apartment is an orientable pseudomanifold, we compute the supremum of the set of $p>1$ such that its top dimensional reduced $\ell^p$-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.