# (Non-)Vanishing results for Lp-cohomology of semisimple Lie groups

For a locally compact, second countable group $$G$$ one can define the continuous $$L^p$$-cohomology $$H^*_{ct}(G,L^p(G))$$ of $$G$$ and the reduced version $$\overline{H}^*_{ct}(G,L^p(G))$$ for all $$p > 1$$. In his influential paper “Asymptotic invariants of infinite groups” Gromov asked if $H^j(G,L^p(G)) = 0$ when $$G$$ is a connected semisimple Lie group and $$j < \mathrm{rk}_{\mathbb{R}}(G)$$. He also remarks that “probably” the $$L^p$$-cohomology in the rank of $$G$$ does not vanish at least for some $$p$$.

The most prominent case $$p=2$$ was already studied by Borel in 1985. He proved the non-vanishing of $$L^2$$-cohomology in a certain range and also established the extremely useful result $\overline{H}^j(G,L^2(G)) = 0 \quad \text{unless}\quad j = \dim(X)/2$ where $$X$$ denotes the symmetric space associated to $$G$$. For general $$p$$ the case $$j = 1$$ has been studied by Pansu and Cornulier-Tessera.

Recently, Marc Bourdon and Bertrand Rémy have been working on this question. They already published first results last year and today appeared a new article on the arXiv. In their first article they prove that $$L^p$$-cohomology is a quasi-isometric invariant. For finitely generated (discrete) groups this was already known by work of Elek. This allows them to transfer the question from semisimple to solvable Lie groups in their new paper. The Iwasawa decomposition $$G = KAN$$ gives rise to an isomorphism $$H^j_{ct}(G,L^p(G)) \cong H^j_{ct}(AN,L^p(AN))$$. The latter can be expressed in terms of $$L^p$$-cohomology à la de Rham. So their recent paper is a study of $$L^p$$ de Rham cohomology of solvable Lie groups. What is the main result? Let $$G$$ be a semisimple real Lie group of rank $$r$$ with finite center. They prove that for all sufficiently large $$p$$

$$H^j_{ct}(G,L^p(G)) = 0$$ for all $$j >r$$

and the reduced $$L^p$$-cohomology does not vanish in degree $$r$$.

The article on the arXiv also raises a question (Question 0.5) which might be of interest to people working in differential geometry:

Question 0.5: Let $$N$$ be a connected simply connected nilpotent Lie group. Let $$X_1,X_2,\dots,X_k$$ be non-trivial left-invariant vector fields on $$N$$. Does N admit a non-zero, compactly supported, $$C^1$$ function $$f : N →R$$, whose integral along every orbit of $$X_i$$ (i = 1,…,k) is null?

They answer the question affirmatively for commuting vector fields and for $$k=2$$.