Thompson's F-group

As piecewise linear homeomorphisms
Thompson's F-group can be defined as the group of all piecewise linear homeomorphisms from the interval $$[0,1]$$ to itself, satisfying the following two conditions:


 * All the break points (points where the slope changes) are dyadic rationals (rationals whose denominator is a power of 2)
 * All the slopes are powers of 2 (they could be 1, or positive, or negative powers)

The group multiplication is simply composition as homeomorphisms.