CAT(0) spaces originated from Riemannian geometry in the 50s as synthetic generalizations of Hadamard manifolds --

simply connected Riemannian manifolds of non-positive sectional curvature.

Already Riemannian geometry provides very interesting examples, like symmetric spaces of non-compact type. However,

the actual universe of CAT(0) spaces is much larger. While it contains many nice non-smooth examples which

carry additional combinatorial structures, like

triangulations or cubulations, mostly coming from group theory, a general CAT(0) space is simply badly singular.

Given this wild zoo of spaces, one inevitably seeks a form of classification.

A CAT(0) space is said to have higher rank if every geodesic lies in a flat plane.

Otherwise it is said to have rank 1.

Ballmann's Rank Rigidity Conjecture classifies CAT(0) spaces X with a cocompact group action by a group G according to their rank.

If X has rank 1, then it displays hyperbolic dynamics: G has many free subgroups, G-periodic geodesics are dense,

the geodesic flow has a dense orbit...

On the other hand, if X has higher rank, then it carries special geometry:

it must be isometric to a symmetric space or a euclidean building or split as a product.

In the talk, I will discuss the conjecture and report on recent progress.

When? | 23.05.2023 17:15 |
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Where? | PER 08 auditoire 2.52 Chemin du Musée 3 1700 Fribourg |

speaker | Prof. Stephan Stadler (MPIM) |

Contact | Département de mathématiques isabella.schmutz@unifr.ch |