If \(M\) is a compact topological manifold, can one say something about its group of homeomorphisms \(\mathrm{Homeo}(M)\)? Of course one can, there is quite a lot to say about it, and there is also quite a lot which we still don’t know. Today I want to mention the following recent result by Csikós-Pyber-Szabó (arXiv:2204.13375): Let … Continue reading "Finite subgroups of homeomorphism groups"

Recently, Guo and Yu pushed the following result to the arXiv (math.KT/2203.15003): For any \(R > 0\) and positive integer \(m\), there exists a constant \(k(R,m)\) such that the following holds. If \((M,g)\) is a Riemannian spin manifold that admits a uniformly bounded, good open cover with Lebesgue number \(R\) and \(R\)-multiplicity \(m\), then \[\inf_{x \in M} \kappa_g(x) \le … Continue reading "A quantitative coarse obstruction to psc-metrics"

Recall that a one-relator group is a group \(G\) admitting a presentation of the form \(G = \langle x_1, \ldots, x_n \ | \ r = 1\rangle\), where \(r\) is a (cyclically reduced) word in the letters \(\{x_1, \ldots, x_n\}\). One-relator groups form an interesting class of groups studied by many people in geometric group … Continue reading "Hyperbolicity of one-relator groups"