Other representations Thompson groups

the thompson group f generated operations on binary trees. here l , t nodes, b , r can replaced more general trees.

the group f has realizations in terms of operations on ordered rooted binary trees, , subgroup of piecewise linear homeomorphisms of unit interval preserve orientation , non-differentiable points dyadic rationals , slopes powers of 2.

the group f can considered acting on unit circle identifying 2 endpoints of unit interval, , group t group of automorphisms of unit circle obtained adding homeomorphism x→x+1/2 mod 1 f. on binary trees corresponds exchanging 2 trees below root. group v obtained t adding discontinuous map fixes points of half-open interval [0,1/2) , exchanges [1/2,3/4) , [3/4,1) in obvious way. on binary trees corresponds exchanging 2 trees below right-hand descendant of root (if exists).

the thompson group f group of order-preserving automorphisms of free jónsson–tarski algebra on 1 generator.