What’s new on the ArXiv: Ricci flow and diffeomorphism groups of 3-manifolds

A new paper proves the contractibility of the space of constant curvature metrics on all 3-manifolds except possibly real projective space. Bamler, Kleiner: Ricci flow and diffeomorphism groups of 3-manifolds, https://arxiv.org/pdf/1712.06197.pdf The Smale conjecture in its original form asserted that the diffeomorphism group of the 3-sphere deformation retracts onto O(3), the isometry group of its … Continue reading "What’s new on the ArXiv: Ricci flow and diffeomorphism groups of 3-manifolds"

What’s new on the ArXiv: A locally hyperbolic 3-manifold that is not hyperbolic

A preprint with a new example shows that the understanding of infinitely generated Kleinian groups will be more complicated than for the finitely generated ones. Cremaschi: A locally hyperbolic 3-manifold that is not hyperbolic, https://arxiv.org/pdf/1711.11568 By the proofs of hyperbolization and tameness, one knows precisely which irreducible 3-manifolds with finitely generated fundamental groups admit hyperbolic … Continue reading "What’s new on the ArXiv: A locally hyperbolic 3-manifold that is not hyperbolic"

What’s new on the ArXiv: Quasi-isometric groups with no common model geometry

Do quasi-isometries between groups always arise from actions on a common model space? Previous counterexamples invoked central extensions of lattices, e.g., of surface groups. A new construction of infinitely many classes is now using amalgams of surface groups. Stark, Woodhouse: Quasi-isometric groups with no common model geometry, https://arxiv.org/pdf/1711.05026.pdf If a group \(\Gamma\) acts geometrically (i.e., … Continue reading "What’s new on the ArXiv: Quasi-isometric groups with no common model geometry"

What‘s new on the ArXiv: A deformation of instanton homology for webs

The four color theorem from graph theory is certainly the most famous problem for which so far only a brute force computational proof exists. A new preprint of Kronheimer-Mrowka supports an approach towards this theorem via homology theories. Kronheimer. Mrowka: A deformation of instanton homology for webs, https://arxiv.org/pdf/1710.05002.pdf The four color theorem says that every … Continue reading "What‘s new on the ArXiv: A deformation of instanton homology for webs"