Venue

Seminarraum C (ground floor) • Mathematikon • Im Neuenheimer Feld 205 • Heidelberg

Every finitely-generated group admits a geometric isometric action on a proper geodesic metric space, which is unique up to quasi-isometry by the Milnor-Schwarz lemma. Similarly, every "finitely-generated approximate group” admits a “geometric quasi-isometric quasi-action” on a proper geodesic metric space which is unique up to quasi-isometry. For many approximate groups one cannot replace “quasi-isometric quasi-action” by “isometric action” in this statement - an obstruction is given by “internal distortion”. Those approximate groups where one can, are called “uniform approximate lattices” - there are very few known non-group examples, which are called “Meyer sets” (a.k.a. mathematical quasi-crystals). These Meyer sets (and conjecturally, all uniform approximate lattices not commensurable to groups) are very rigid. For example, they show super-rigidity like phenomena even in rank one, e.g. approximate surface groups have trivial Teichmüller space. Conjecturally, some form of super-rigidity should hold for all uniform approximate lattices which aren’t groups (even in rank one). I will explain what is known (based on joint work with Michael Björklund, Matt Cordes and Vera Tonic) and also all words in quotation marks.
Recently, we observed (with Fanny Kassel and Jean-Philippe Burelle) that there should be a rigidity phenomenon for hyperbolic approximate groups which is even stronger than superrigidity. We believe that every convex-cocompact isometric action of a hyperbolic approximate group on a rank one-symmetric space is essentially a uniform approximate lattice embedding into its Zariski closure, which is reductive - unless the approximate group is commensurable to a group. In particular, the limit set has integral dimension and conjecturally we get super-rigidity from convex-cocompactness alone. The natural context of these results seems to be that of (non-existence of) Anosov representations of hyperbolic approximate groups.

A theorem of Katok implies that any aperiodic flow on a compact 3-manifold has zero topological entropy. Aperiodic flows can be constructed using the Krystyna Kuperberg's plug. I will present a one-parameter family of plugs, containing the Kuperberg plug, plugs with simple dynamics (in the sense that the maximal invariant set is a cylinder) and plugs whose flow has positive topological entropy.

We introduce new invariants of topological spaces: the p-adic Betti numbers and the p-adic torsion. We show that they are p-adic limits of classical topological invariants and we give a couple of examples. The p-adic invariants contain information about cohomology growth and we will see that in this respect they behave like antagonists of L2-invariants.