  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 (also known as V) 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 plynomial time). We show that
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.
