Course Announcemment - Hyperbolic groups - MTG 5376 - Fall/2006
Instructor - Sergio Fenley
  
  
        This will be a course on the geometry of groups, that is,
viewing groups as geometric objects, where the large scale geometry
plays an essential role.
        We start with the Cayley graph: this is a one dimensional complex,
whose zero complex is the original group and edges correspond to generators
in a fixed presentation of the group. There is an associated metric for the
group. The important objects are the large scale or asymptotic objects,
which are invariant by what is called a quasi-isometry. This is
a map which allows for distances to be distorted a bounded
multiplicative amount. Amazingly enough there is an enormous amount
of information obtained even with this flexibility.
        We will cover the fundamental theorem of Geometric Group
theory and its implications.
        A substantial part of the course will be devoted to hyperbolic groups.
This type is extremely common and has fundamental properties. We will
discuss several equivalent definitions: 1) Inner product, 2) Thin triangles
(in its various forms), 3) Isoperimetric inequalities. Topics will
include: approximation by trees, geodesics and quasigeodesics,
stability of quasigeodesics, ideal boundary and homeomorphisms
and related topics.
  
        Geometric group theory is a subject which (as a defined area in
mathematics) is relatively new - it is 20-25 years old. It is growing   
tremendously and currently there are several important branches. One     
main thrust was given by Gromov in his seminal article in 85, entitled
Hyperbolic groups.