---
tag: VK50GVVFTN
kind: object
name: Thompson groups
msc: ["20F65", "20E32", "20F05"]
instance_of: [TJKANTKASZ]
sources:
  - url: https://en.wikipedia.org/wiki/Thompson_groups
same_as:
  wikidata: Q4150823
  wikipedia: Thompson groups
---

## Description

Thompson's group $F$ is the [[group|G5S1M9]], under composition, of all [[homeomorphisms|0QT00PB5G2]] $f \colon [0, 1] \to [0, 1]$ such that:

1. $f$ is piecewise linear, with finitely many break points;
2. every break point is a dyadic rational, a number $k / 2^j$ with integers $k$ and $j \ge 0$;
3. every slope of $f$ is an integer power of $2$.

Thompson's group $T$ is the group of such piecewise linear maps of the circle $[0, 1] / \set{0, 1}$, which also map the dyadic rationals onto themselves.
Thompson's group $V$ is the group of right-continuous bijections of $[0, 1)$ that are piecewise linear in the same sense.
They may have finitely many jumps, all at dyadic rationals.
The inclusions $F \subset T \subset V$ hold.

For example, the map $A$ with $A(x) = x / 2$ on $[0, 1/2]$, $A(x) = x - 1/4$ on $[1/2, 3/4]$ and $A(x) = 2x - 1$ on $[3/4, 1]$ lies in $F$.
The pieces agree at the break points: $1/2 \mapsto 1/4$ and $3/4 \mapsto 1/2$, and the slopes are $1/2$, $1$ and $2$.

## Remarks

All three groups are [[finitely presented|TJKANTKASZ]].
$F$ has the infinite presentation $\langle x_0, x_1, x_2, \ldots \mid x_i^{-1} x_n x_i = x_{n+1} \text{ for } i < n \rangle$, and it is generated by $x_0$ and $x_1$ with two relations.
The commutator subgroup $[F, F]$ is simple, and $F / [F, F] \cong \Z^2$.
$F$ is torsion-free and contains no [[free subgroup|0KTW26C80J]] of rank $2$.
Whether $F$ is [[amenable|CEAT9BESKK]] is a well-known open problem.
$T$ and $V$ are infinite [[simple groups|0YMYMNQ0QZ]] that are finitely presented; they were the first known examples of such groups.

## Explanations

### L1 human

Cut the interval $[0, 1]$ into pieces by repeatedly halving, and do the same with another copy of $[0, 1]$, with the same number of pieces.
Stretch each piece of the first copy linearly onto the matching piece of the second.
The maps obtained this way form Thompson's group $F$.
The groups $T$ and $V$ allow the pieces to be rotated cyclically or shuffled in any order.
These simple-looking groups answer many questions in group theory and serve as counterexamples to many guesses.