Abstract
We prove new results about the remarkable infinite simple groups
introduced by Richard Thompson in the 1960s. We give a faithful
representation in the Cuntz C⋆-algebra. For the finitely
presented simple group V we show that the word-length and the table
size satisfy an n log n relation. We show that the word problem of
V belongs to the parallel complexity class AC1 (a subclass of
P), whereas the generalized word problem of V is undecidable.
We study the distortion functions of V and show that V 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 V, the set of Dehn functions of
finitely presented groups, and the set of time complexity functions of
nondeterministic Turing machines.