Thompson groups

Description

Thompson's group $F$ is the group, under composition, of all homeomorphisms $f \colon [0, 1] \to [0, 1]$ 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 $2$.