Many classical group properties imply relations of special form for generators of groups. For example, if a word in generators represents a nontrivial element of a simple group, then each generator can be represented as a product of words (and their inverses) conjugated to the given word. If a group is periodic, then some power of every word in this group is equal to the identity element. So, groups with prescribed algebraic properties can, inversely, be constructed by adding relations so that in every step we obtain a group with finite set of relations, while the final group is a limit of these finitely generated groups. An induction of such type can be under control, if all groups arising on each step have hyperbolic geometry under word metric.
The goal of the course of lectures is to show
how special metric properties of finitely presented groups allow to
provide prescribed algebraic characteristics of limit groups. On
practical seminars we shall use theoretical background to construct
groups with prescribed properties.
This course will consist of eight lectures. The
particioants are recommended to read the following monography:
A.Yu.Ol'shansky, Geometry of defining relations in groups. Translated
from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its
Applications (Soviet Series), 70. Kluwer Academic Publishers Group,
Dordrecht, 1991. xxvi+505 pp. ISBN: 0-7923-1394-1.