(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"

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"

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"

Online Fernklausuren in Mathematik

Im letzten Semester mussten aufgrund der Pandemie viele Prüfungen in anderen (oft digitalen) Formaten abgehalten werden. Für das laufende Sommersemester – auch wenn sich die Lage derzeit bessert – scheint es mir wahrscheinlich, dass zumindest Prüfungen mit vielen Teilnehmern weiterhin Einschränkungen unterworfen sind. Daher will ich die Gelegenheit nutzen, um ein paar Überlegungen zu sammeln, … Continue reading "Online Fernklausuren in Mathematik"

Topological complexity

Rummaging in the arXiv I ran across this article and the notion of topological complexity which is really appealing. The idea of topological complexity isn’t quite new, it was developed by M. Farber in a short article published in 2003 in Discrete & Computational Geometry. The article doesn’t even have a proper review on MathSciNet, … Continue reading "Topological complexity"