Event Details

Author: Maciej Czarnecki (University of Lodz, Poland)

The aim of the lectures is to provide a brief but extensive introduction to the theory metric spaces of nonpositive curvature and its applications in the geometric group theory.
Contents of lectures:

  • Classical hyperbolic geometry: models and metric properties. We shall describe geometry of the real hyperbolic spaces including distances in different models, geodesics, trigonometry and ideal boundary.
  • Group actions: isometries, reflections, quasi-isometries and group graphs. We shall start from Euclidean and hyperbolic isometries and their reflection subgroups to provide examples of group actions on metric spaces and their properties. Then using a more flexible notion of quasi-isometry we shall study relationships between a metric space and a group acting on it. On the other hand, we shall observe metric properties of graphs for finitely generated groups.
  • CAT(0) spaces: geometry, isometries and boundary. Here we shall describe CAT(0) space as those having geodesic triangles "thinner" than Euclidean ones. Such spaces, even if a CAT(0) condition is satisfied locally, have many properties similar to nonpositively curved Riemannian manifolds. In particular, we could build an ideal boundary in the universal cover and act by isometries on the space as well as on this boundary.
  • Hyperbolic groups: algebra vs. geometry through natural algebraic problems.A metric space is hyperbolic if geodesic triangles are uniformly thin. This makes possible to compare its properties to those possesed by the real hyperbolic space. We shall present some results on groups having hyperbolic Cayley graphs which provide a bridge between their geometry and purely algebraic questions.

    • JUEVES 10 y VIERNES 11: 9:00h a 12:00h