2002/04/24 by Jean-Camille Birget, Birget, Jean-Camille · 1 citation
Computer Science · Mathematics · #20F10 #68Q15 #Computability, Logic, AI Algorithms #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:20F10 #msc:68Q15 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0204292
openalex publication_date 2002/04/24 · arxiv created 2003/10/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Tfin (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 Tfin belongs to the parallel complexity class AC1 (a subclass of plynomial time). We show that the generalized word problem of Tfin is undecidable. We study the distortion functions of Tfin and we show that Tfin 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 Tfin, the set of all Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.