Branched coverings of symmetric spaces and nonpositive curvature

• Applicant: Dr. Stephan Stadler
• Subject Area: Mathematics
• Term: from 2016 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 329206558

Branched coverings produce metric spaces of nonpositive curvature with intriguing features. For instance, I proved in my thesis that a certain 4-manifold which allows for a branched covering over a product of hyperbolic surfaces and therefore is nonpositively curved in the metric sense is not smoothable, i.e. does not carry a Riemannian metric of nonpositive curvature. This answered a question of Gromov from the year 1985. Refining and completing the techniques from my thesis makes it possible to prove rigidity theorems for total spaces of branched coverings of symmetric spaces. In particular one obtains information about the quasi-isometry groups of their universal coverings. Examples show that these groups are not rigid in the strict sense, i.e. they are not virtually isomorphic to the respective isometry groups. The aim of this project is to reveala form of semi-rigidity which virtually identifies the quasi-isometry group of a branched covering with the quasi-isometry group its ramification locus.

• DFG Programme: Research Fellowships
• International Connection: United Kingdom, USA
• Participating Institution: New York University ; Courant Institute of Mathematical Sciences
• Host: Professor Bruce Kleiner, Ph.D.