Non-vanishing for group $L^p$-cohomology of solvable and semisimple Lie groups

TL;DR

This paper proves that the $L^p$-cohomology of certain Lie groups does not vanish at the group's rank for large p, confirming an optimal formulation of Gromov's vanishing question and employing spectral sequences and invariance techniques.

Contribution

It establishes non-vanishing results for $L^p$-cohomology of semisimple and solvable Lie groups at the rank, using spectral sequences and comparison theorems.

Findings

Non-vanishing of $L^p$-cohomology at the rank for large p

Confirmation of Gromov's vanishing question as optimal

Use of spectral sequences and quasi-isometry invariance

Abstract

We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov's question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.