Thompson's F-group

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Definition

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.