### An implication of the Farrell-Jones conjecture

A ‘well-known’ implication of the Farrell-Jones conjecture (for a given group G) is that the map $\widetilde{K_0(\mathbb{Z}G)} \to \widetilde{K_0(\mathbb{Q}G)}$ in reduced algebraic K-theory is rationally trivial. What at first might seem as a technical statement about algebraic K-theory turns out to have an interesting geometric consequence. It implies the Bass conjecture, which is equivalent to … Continue reading "An implication of the Farrell-Jones conjecture"

### Kaplansky’s direct finiteness conjecture

Not too long ago I blogged about the first counter-example to Kaplansky’s unit conjecture (link) stating that there are no non-trivial units in the group ring K[G] for K a field and G a torsion-free group. A related conjecture of Kaplansky (one that I was not aware of until recently) is that K[G] is directly … Continue reading "Kaplansky’s direct finiteness conjecture"

### Message from the EMS president

In the last EMS Magazine (2021/No. 121) Volker Mehrmann reflected in his editorial (link) on the bygone (virtual) European Congress 8ECM. At the end he asked to write to him our opinions about the matters that he addressed, which I did. I want to share here now my e-mail to him with you: Lieber Volker, … Continue reading "Message from the EMS president"

### Lie groups acting on countable sets

Does every connected Lie group act faithfully on a countable set? In other words: is every Lie group a subgroup of $$\mathrm{Sym}(\mathbb{N})$$? This question is sometimes called Ulam’s problem and there is recent progress in a paper of Nicolas Monod. Monod proves that every nilpotent connected Lie group acts faithfully on a countable set. It … Continue reading "Lie groups acting on countable sets"

### Topological CAT(0)-manifolds

It is an interesting and important fact that a contractible manifold (without boundary) is not necessarily homeomorphic to Euclidean space. This makes the classical Cartan-Hadamard theorem, stating that a contractible manifold equipped with a Riemannian metric of non-positive sectional curvature is diffeomorphic to Euclidean space, even more powerful. One can ask now whether one can … Continue reading "Topological CAT(0)-manifolds"

### PSC obstructions via infinite width and index theory

In a recent preprint (arXiv:2108.08506), Yosuke Kubota proved an intriguing new result on the relation of largeness properties of spin manifolds and index-theoretic obstructions to positive scalar curvature (psc): Let $$M$$ be a closed spin $$n$$-manifold. If $$M$$ has infinite $$\mathcal{KO}$$-width, then its Rosenberg index $$\alpha(M) \in \mathrm{KO}_n(\mathrm{C}^\ast_{\max} \pi_1 M)$$ does not vanish. Let us … Continue reading "PSC obstructions via infinite width and index theory"

### (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"