Bass-Serre theory describes group splittings as actions on trees and reconstructs a graph of groups from the stabilizers of such an action.
For , the Bass-Serre tree has vertices and edges , with joining to . Vertex degrees are the indices of the corresponding images of .
Given isomorphic subgroups and an isomorphism , the HNN extension is
(1)
Its stable letter records the loop edge in the associated graph of groups.
The Baumslag-Solitar group is
(1)
In , an alternating product of elements of and is nonidentity. More generally, choosing coset representatives gives a unique reduced normal form.
The fundamental group of the Klein bottle has presentation
(1)
Its subgroup has index two.

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Bass–Serre theory by Wikipedia Bot 1
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Bass–Serre theory is a branch of algebraic topology that studies the relationships between groups and their actions on trees (in a combinatorial sense). Developed by mathematicians Hyman Bass and Jean-Pierre Serre in the 1960s, the theory provides a framework for understanding certain types of groups, particularly finitely generated groups that can be decomposed in terms of simpler pieces.
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