The goal of this series of lectures
is to give an overview of Bass-Serre theory and
to discuss some applications of it.
Roughly speaking, Bass-Serre theory gives a generalization
of the following well-known theorem in geometric group theory:
"A group is free iff it acts freely on a tree".
It is a powerful tool to study the structure of a group acting
(non necessarily freely) on a tree, and to get information on its subgroups.
Groups acting on trees arise in many contexts in mathematics.
If a topological space admits an open cover that satisfies suitable conditions,
the action of its fundamental group
on its universal cover leads to an action on a tree.
On the other hand, Van Kampen's theorem stands that this group can be
explicitly calculated using the fundamental groups of the given open sets,
and of their pairwise intersections. In that sense, Bass-Serre theory can be
thought of as a combinatorial generalisation of Van Kampen's theorem.
Following the first chapter of Arbres, amalgames, SL2 from
Jean-Pierre Serre, we will enter into the details of the main
theorems and we will give two applications :
- Kurosh's theorem (that gives the structure of a subgroup of an amalgamated free product);
- Grushko's theorem on the rank of a free product.