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 “round” constant curvature metric. It was proved by Hatcher in 1983. Also for hyperbolic 3-manifolds it was proven by Gabai in 2001 that the inclusion of the isometry group into the diffeomorphism group is a homotopy equivalence. Together with other known results for e.g. lens spaces, this gave evidence for the generalized Smale conjecture, asserting that the inclusion of the isometry group into the diffeomorphism group should be a homotopy equivalence for all 3-manifolds of non-zero constant curvature.

The natural action of the diffeomorphism group on the space of constant curvature metrics is transitive (because of rigidity of constant curvature metrics) and defines a fibration with the isometry groups as its fiber. Thus, in view of the long exact homotopy sequence, the weak homotopy equivalence of Isom(X)–>Diff(X) is equivalent to weak contractibility of the space of constant curvature metrics. Because all involved spaces have the homotopy type of CW-complexes, one can replace “weak homotopy equivalence” by “homotopy equivalence” and “weak contractibility” by “contractibility” in this equivalence.

Thus the generalized Smale conjecture reduces to showing that the space of constant curvature metrics has trivial homotopy groups, which this paper proves for all 3-manifolds except the 3-sphere and the real projective space. (And the case of the 3-sphere is of course known from Hatcher.)

The new preprint attacks this problem via Ricci flow. The naive idea to prove vanishing of homotopy groups would be to take a sphere in the space of constant curvature metrics, which (by contractibility of the space of all Riemannian metrics) will bound a ball in the space of Riemannian metrics, and then use Ricci flow to deform this ball into a ball in the space of constant curvature metrics. Of course, this does not work that easily because the Ricci flow will develop singularities along the flow. The main part of the paper consists in analyzing these singularities and showing that the constant curvature metric can be extended over neighborhoods of the singularities.

As the authors point out, their proof also provides an alternative argument for Moscow rigidity. Namely, it shows that the space of constant curvature metrics modulo Isom(X) is connected, which together with the known finiteness implies that it consists of one point only.

]]>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 metrics: they have to be atoroidal and have infinite fundamental group.

The study of hyperbolic 3-manifolds with infinitely generated fundamental group is more involved. For example there are Cantor sets in the 3-sphere such that the complement admits a complete hyperbolic metric. (J. South, M. Store: A Cantor set with hyperbolic complement, Conformal Geometry and Dynamics 17, 58-67, 2013.)

For irreducible 3-manifolds with infinitely generated fundamental group one has two necessary conditions for hyperbolization:

* has to be divisible, i.e., for every there are infinitely many (distinct) such that for some $n$, and

* has to be locally hyperbolic, i.e., every cover with finitely generated fundamental group has to be hyperbolic.

One may wonder whether these conditions are already sufficient or whether there exist 3-manifolds satisfying both conditions without being hyperbolic. (This question is attributed to Agol.)

The new paper shows that such a manifold indeed exists. The example is the thickening of the 2-complex obtained from gluing to an infinite annulus countably many copies of a genus 2 surface such that each is glued to . The surfaces cobound a hyperbolizable manifold, which can be used to prove that is locally hyperbolic. On the other hand, the infinite annulus is used to show that cannot be hyperbolic.

]]>Main event of the meeting were the lectures of Oscar Randal-Williams from Oxford, who discussed work on the cohomology of the mapping class group beyond the stable range.

Some more impressions:

In any case, if you‘d like to see the ceremony, the math part starts at 1:22:30.

The breakthrough prize for 2018 was given to Christopher Hacon and James McKernan for their contributions to the birational classification of higher-dimensional algebraic varieties.

The New Horizons Prizes went to Aaron Naber, Maryna Viazovska, Zhiwei Yun and Wei Zhang.

Stark, Woodhouse: Quasi-isometric groups with no common model geometry,

https://arxiv.org/pdf/1711.05026.pdf

If a group acts geometrically (i.e., properly and cocompactly) on a space $X$, then it is quasi-isometric to $X$ by the Milnor-Svarc theorem. Thus, a standard way of proving two groups to be quasi-isometric is to let them act geometrically on the same space. For example, hyperbolic space in dimensions admits many geometric actions by non-commensurable groups which are thus quasi-isometric.

One may ask for the converse: do quasi-isometries between groups always arise from geometric action on a common model space. (Note that groups always act geometrically on some space, namely their Cayley graph. However, the quasi-isometry between the Cayley graphs of two quasi-isometric groups only serves to promote an action on one Cayley graph to a quasi-action on the other and it is not always the case that a quasi-action can be promoted to an actual action.)

Known examples of quasi-isometric groups not having a common model geometry arise as groups of the form

where is a uniform lattice in a simple Lie group with .

If $G$ has property $T$, then also does, while

does not, thus the two groups can not be quasi-isometric.

This argument does not apply to , i.e., for surface groups , but in this case another argument applies. Two groups are called measure equivalent if they have a (not necessarily cocompact) proper action with finite-volume fundamental domains on some measure space. Such a measure equivalence defines a coccycle by figuring out in which copy of the fundamental domain an element with lies. In the case of with $latex $\Gamma\subset SL(2,{\mathbb R})$ it turns out that the groups are measure equivalent but the cocycle $\alpha$ is not -integrable. This, however, would necessarily be the case if the quasi-isometry would come from a cocompact action on a common model space. (Das, Tessera: Integrable measure equivalence and the central extension of surface groups, https://arxiv.org/pdf/1405.2667)

The examples in the new construction are of a different type. They are fundamental groups of amalgams of surfaces, i.e., of spaces obtained from $k$ compact surfaces with one boundary component by identifying all the boundary components. These fundamental groups are known to be quasi-isometric to each other (Malone, Topics in Geometric Group Theory, PhD-thesis, University of Utah, 2011) and for k=4 there is a classification up to commensurability (Stark, Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams, Geometrie Dedicata, 186, 39–74, 2017) which in particular implies that there are infinitely many commensurability classes. The new paper shows that the non-commensurable groups can not act geometrically on the same space.

Very roughly, the idea of the proof is the following. If two groups act on the same model space, then they have the same boundary at infinity. From the boundary at infinity one can reconstruct the JSJ-tree of the JSJ-decomposition. The JSJ-tree can be used to promote the model space to a CAT(0) cube complex, on which both groups act geometrically. While the groups need not be finite index in the isometry group of the cube complex, It turns out that there is a certain subgroup of this isometry group in which both groups happen to be finite index subgroups. This proves commensurability.

]]>Michael Rios and David Chester in two videos try to explain the essence of the new work and, for example, the compatibility with Garrett Lise’s E8 theory. The video is remarkable not only for the content but also for the form: not a panel lecture, but a campus walk.

]]>Kronheimer. Mrowka: A deformation of instanton homology for webs, https://arxiv.org/pdf/1710.05002.pdf

The four color theorem says that every planar map can be colored by four colors such that adjacent countries correspond to different colors.

It is known from Tait‘s work in the 19th century that the four color theorem is equivalent to the existence of Tait colorings on bridgeless trivalent graphs in the plane. Here a Tait coloring is a coloring of the edges by 3 colors such that each vertex is adjacent to 3 edges of different colors. A bridgeless graph is a connected graph which can not be disconnected by removing only one edge.

In an earlier paper, the authors used a variant of the instanton homology for knots to define an invariant of trivalent graphs in (which they call „webs“). They prove that it is non-zero for bridgeless graphs, and they conjecture that it is equal to the number of Tait colorings of the graph. (Kronheimer, Mrowka: „Tait colorings, and an instanton homology for webs and foams“, https://arxiv.org/abs/1508.07205)

In the new paper, the authors construct a deformation of their homology theory for which they can actually show that its dimension agrees with the number of Tait colorings. The dimension of the deformed homology is at most that of the original homology theory. Of course, to prove the four color theorem one would need the opposite inequality.

]]>The last of this years lectures has been given by Cédric Villani at October 23. It was a big success with more than 350 listeners in the theatre of Regensburg, nicely located within the old town.

When organizer Bernd Ammann approached the speaker in early 2016, he could not guess that Villani would be a busy politician in 2017. Villani, who was elected to the Assemblée Nationale this summer, is now accountable for the plans to promote the development of artificial intelligence. And so he used the opportunity of his trip to Regensburg to discuss the subject in an organized round with local experts and leaders.

The public talk, however, was about mathematics. The problems of artificial intelligence only showed up in magician Thomas Fraps’ pre-programme, where a well-known online retailer and a well-known search engine were playing a role in some of the magic tricks.

Villani’s talk was “about triangles, gazes, prices and men”, according to Villani the story of the encounter of three a priori unrelated fields: non-Euclidean geometry, theory of gazes, and economics. (Or in mathematical terms: Ricci curvature, entropy, and optimal transport.)

All pictures below are Copyright by Raphael Zentner. The organizers have a website about the event.

Manuel Amann receives the von-Kaven-Preis from Christian Bär.

Magician Thomas Fraps …

… plays some cardgame with the prizewinner …

… and explains it mathematically.

Google knows the answer.

Organizer Bernd Ammann introduces Cédric Villani.

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The list of the talks can be found here.

Interesting from the point of view „Geometry at Infinity“ was especially the talk of Olivier Guichard regarding work of Kapovich-Leeb-Porti and Labourie on convex-cocompact groups in higher rank symmetric spaces:

Another talk was by Serge Cantat about the recent progress towards Zimmer‘s conjecture.

The other two talks were about representation theory.

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