Isoperimetric and isodiametric functions of groups

Abstract

This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. In particular, we show how to construct a finitely presented group with NP-complete word problem. One of the main results of this paper states that if a real number α ≥ 4 is computable in time ≤ 22Cm for some constant C > 0 then nα is equivalent ("big O") to the Dehn function of a finitely presented group. The smallest isodiametrie function of this group is n3α/4. On the other hand, if nα is equivalent to the Dehn function of a finitely presented group then a is computable in time ≤ 222Cm for some constant C. Being computable in time T(n) means that there exists a Turing machine which, given n, computes a binary rational approximation of α with error at most 1/2n+1 in time at most T(n). This implies that, say, functions nπ+1, ne2 and nα for all rational numbers α ≥ 4 are equivalent to Dehn functions of some finitely presented groups and that nπ and nα for all rational numbers α ≥ 3 are equivalent to the smallest isodiametrie functions of finitely presented groups. Moreover, we describe all Dehn functions of finitely presented groups &sp; n4 as time functions of Turing machines modulo two conjectures: 1. Every Dehn function is equivalent to a superadditive function. 2. The square root of the time function of a Turing machine is equivalent to the time function of a Turing machine.