---
tag: 6M731X66RD
kind: concept
name: Bass–Serre theory
msc: ["20E08", "20E06"]
sources:
  - url: https://en.wikipedia.org/wiki/Bass–Serre_theory
same_as:
  wikidata: Q4868208
  wikipedia: Bass–Serre theory
---

## Summary

Bass–Serre theory describes the groups that act on [[trees|062D9KXJB1]] by building them from simpler groups.
Its central object is a [[graph of groups|DC1CQMD3WK]]: a graph with a group at each vertex and each edge.
The structure theorem says that a group acting on a tree without inversions is the fundamental group of the quotient graph of groups.
The vertex and edge groups of this graph of groups are stabilisers.
Conversely, the fundamental group of a graph of groups acts on a tree, the Bass–Serre tree.
It unifies [[free products|TM7QS7P7X9]], amalgamated free products and [[HNN extensions|YEYERRY37K]].
It gives short proofs of subgroup theorems such as the [[Nielsen–Schreier theorem|W97X7WX5G8]] and the [[Kurosh subgroup theorem|RY8K5RQJSR]].

## Explanations

### L1 human

A tree is a graph with no cycles, so there is exactly one path between any two points.
When a group moves a tree around by symmetries, the quotient picture is a graph, and each vertex and edge of it remembers the group of symmetries fixing it.
Bass–Serre theory says one can rebuild the whole group from this decorated graph.
For instance, $\mathrm{SL}_2(\Z)$ acts on a tree with quotient a single edge.
This shows that it is an amalgam of a cyclic group of order $4$ and a cyclic group of order $6$ over a common subgroup of order $2$.
A subgroup of the group still acts on the same tree, which is why the theory proves so much about subgroups.