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

### New book about Freedman’s proof

Today I learnt from an article in the QuantaMagazine (link to article) that there is finally a new book trying to explain Freedman’s proof of the 4-dimensional Poincaré conjecture (link to book). The article is fun to read since it contains statements of the involved people about how the whole ‘situation’ about the non-understandable write-up … Continue reading "New book about Freedman’s proof"

### Loewner’s “forgotten” theorem

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"

### Rigidity implication of the Novikov conjecture

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"

### Space of psc-metrics on non-spin manifolds

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"

### A question about the first $$L^2$$-Betti number

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"

### Computer assisted verification of contemporary mathematics

Half a year ago Steffen Kionke wrote a blog post on condensed mathematics wherein he mentioned that Peter Scholze put up a challenge to formally verify a key fundamental result of a joint paper with Clausen: a result Scholze terms his most important theorem to date. Today I want to inform you that the mentioned … Continue reading "Computer assisted verification of contemporary mathematics"

### A hyperbolic 5-manifold which fibres over the circle and subgroups of hyperbolic groups

Recently appeared a highly interesting paper by Italiano, Martelli, Migliorini on the arXiv (2105.14795) (Okay, this was already three weeks ago – I don’t follow the “geometric topology” – and someone had to point me there). The paper solves at least two longstanding open problems by studying an explicit 5-dimensional hyperbolic manifold. It is well-known … Continue reading "A hyperbolic 5-manifold which fibres over the circle and subgroups of hyperbolic groups"