THE GROUPS OF RICHARD THOMPSON AND COMPLEXITY

JEAN-CAMILLE BIRGET
2004 International journal of algebra and computation  
We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We define the groups as partial transformation groups and we give a faithful representation in the Cuntz C ⋆ -algebra. For the finitely presented simple group T fin we show that the word-length and the table size satisfy an n log n relation, just like the symmetric groups. We show that the word problem of T fin belongs to the parallel complexity class AC 1 (a subclass of P). We show
more » ... the generalized word problem of T fin is undecidable. We study the distortion functions of T fin and we show that T fin contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of T fin , the set of all Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.
doi:10.1142/s0218196704001980 fatcat:gsztws2lbfdobcldqh7ndu6wmq