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 \(j < \mathrm{rk}_{\mathbb{R}}(G)\). … Continue reading "(Non-)Vanishing results for Lp-cohomology of semisimple Lie groups"

Yesterday I discovered an instructive paper on “Loewner’s forgotten theorem” by Peter Albers and Serge Tabachnikov on the arXiv. I don’t know in how far the result has really been forgotten, it was published in the Annals in 1948. Fair enough, it has only 4 citations on MathSciNet. So what is Loewner’s theorem? It says … Continue reading "Loewner’s “forgotten” theorem"

Part of my own research is related to the Novikov conjecture, and hence I am always interested to see applications of it. When reading the preprint Essentiality and simplicial volume of manifolds fibered over spheres by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892) I saw another one of these applications (Theorem D therein). I do not … Continue reading "Rigidity implication of the Novikov conjecture"

Recently I was having a look at the preprint Essentiality and simplicial volume of manifolds fibered over spheres by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892). In Theorem C therein they construct a closed, totally non-spin manifold (i.e., one whose universal cover is even not spin) whose space of Riemannian metrics of positive scalar curvature has … Continue reading "Space of psc-metrics on non-spin manifolds"

In a recent arXiv preprint (arxiv:2106.15750) J. A. Hillman discusses a new homological approach towards some old results on 3-manifold groups due to Elkalla. In his article he runs into an interesting question concerning the first \(L^2\)-Betti number of finitely generated groups: Assume that a finitely generated group G has infinite subgroups \(N\leq U\) such … Continue reading "A question about the first \(L^2\)-Betti number"