Geometric Analysis 15-Math-8002 Gromov Hyperbolicity

Department of Mathematical Sciences

This page is a work in progress! All information is subject to change (Last revised 23 January 2015)

Basic Course Description

Geodesic triangles in the hyperbolic plane have the property that every point on a given side is at most distance two from one of the other sides, regardless of how big the triangle is. A Gromov hyperbolic metric space is a metric space with this thin triangles property. Amazingly, this simple definition leads to deep mathematics, particularly in relation to group theory. Gromov hyperbolic groups have good algebraic properties (for example, solvable word problem) and are "generic": if you pick a group at random, it will be Gromov hyperbolic.

Hyperbolic space \(\mathbb{H}^{n+1}\) has an $n$-sphere \(\mathbb{S}^{n}\) at infinity which has a conformal structure. This was used, for example, in the original proof of Mostow rigidity. Likewise, a Gromov hyperbolic space \(X\) has a boundary at infinity \(\partial_\infty X\) which carries a canonical topological (in fact, conformal) structure. However, unlike the usual sphere, this boundary will often have fractal like properties.

The aim of this course is to give an introduction to Gromov hyperbolic metric spaces and to explore some of the connections between the geometry of the boundary at infinity and quasiconformal function theory.

Here is a preliminary outline of topics that we intend to cover.

• Definition of Gromov hyperbolic groups and some basic properties
• Boundaries of Gromov hyperbolic groups and visual metrics
• Quasi-conformal and quasi-Mobius maps on metric spaces

References

• Books:
  • Mikhail Gromov, Hyperbolic Groups, pp.75-263, Essays in group theory, Springer, 1987.
  • Mario Bonk, Juha Heinonen, and Pekka Koskela, Uniformizing Gromov hyperbolic Spaces, 2001.
  • Martin Bridson and André Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, 1999.
  • Dmitri Burago, Yuri Burago, and Sergei Ivanov, A Course in Metric Geometry, 2004.
  • M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et Théorie des Groupes, Springer-Verlag, 1990.
  • Étienne Ghys and and Pierre de la Harpe, P. (editors), Sur les Groupes Hyperboliques d'après Mikhael Gromov, Birkhäuser Boston Inc., 1990.
  • Pierre be la Harpe, Topics in Geometric Group Theory, University of Chicago Press, 2000.
    Lots of interesting examples.
  • John Roe, Lectures on Coarse Geometry, AMS, 2003.
  • H. Short (editor), and B. Bowditch, Notes in Group Theory from a Geometrical Viewpoint, World Scientific Publishing Co., Inc., 1991.
  • Brian Bowditch, A Course on Geometric Group Theory, Mathematical Society of Japan, 2006.
    Also available at his website.
• Other references:
  • The 2006 ICM survey paper by Bruce Kleiner, The asymptotic geometry of negatively curved spaces: uniformization, geometrization, and rigidity, pdf.
  • There is a blog/textbook of Henry Wilton's online course at UT Austin from Spring 2009 here.