Free group by suri aisiit.pdf - Free groups basics B.Sury Stat-Math Unit Indian Statistical Institute Bangalore India AIS May 2010 IIT Bombay \u201cOnly

# Free group by suri aisiit.pdf - Free groups basics B.Sury...

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Free groups - basics B.Sury Stat-Math Unit Indian Statistical Institute Bangalore, India AIS - May 2010 - IIT Bombay “Only that thing is free which exists by the necessities of its own nature, and is determined in its actions by itself alone” - Baruch Spinoza “Everything that is really great and inspiring is created by the individual who can labor in freedom” - Albert Einstein “Now go we in content To liberty, and not to banishment.” ... W.Shakespeare (As You Like It) “Freedom is like drink. If you take any at all, you might as well take enough to make you happy for a while” - Finley Peter Dunne 1
Introduction Free objects in a category (whatever these animals may be) are the most basic objects in mathematics. A paradigm is the theory of free groups. They arose naturally while studying the geometry of hyperbolic groups but their fundamental role in group theory was recognized by Nielsen (who named them so), Dehn and others. In these lectures, we introduce free groups and their subgroups and study their basic properties. The lectures by Professor Anandavardhanan would treat the more general notions of free products with amalgamation and HNN extensions in detail. However, it is beneficial to look at them already for our purposes as they bear repetition. We shall do so briefly in the course of these lectures. The notions of free groups, free products, and of free products with amalga- mation come naturally from topology. For instance, the fundamental group of the union of two path-connected topological spaces joined at a single point is isomorphic to the so-called free product of the individual fundamental groups. (More generally) The Seifert-van Kampen theorem asserts that if X = V W is a union of path-connected spaces with V W non-empty and path- connected, and if the homomorphisms π 1 ( V W ) π 1 ( V ) and π 1 ( V W ) π 1 ( W ) induced by inclusions, are injective, then π 1 ( X ) is isomorphic to the so-called free product of π 1 ( V ) and π 1 ( W ) amalgamated along π 1 ( V W ). Many naturally occurring groups can be viewed in terms of these construc- tions. For instance, SL (2 , Z ) is the free product of Z / 4 and Z / 6 amalgamated along a subgroup isomorphic to Z / 2 . The fundamental group of the Klein bottle is isomorphic to the free product of two copies of Z amalgamated along 2 Z . The so-called HNN extensions also have topological interpretations. Suppose V and W are open, path-connected subspaces of a path-connected space X and suppose that there is a homeomorphism between V and W inducing isomorphic embeddings of π 1 ( V ) and π 1 ( W ) in π 1 ( X ). One constructs a space Y by attaching the handle V × [0 , 1] to X , identifying V × { 0 } with V and V × { 1 } with W . Then, the fundamental group π 1 ( Y ) of Y is the HNN extension of π 1 ( V ) relative to the isomorphism between its subgroups π 1 ( V ) and π 1 ( W ).