Isoperimetric and Isodiametric Functions of Groups

This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C22Cm for some constant C > 0 then nα is equivalent (“big O”) to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is n3/4α. On the other hand if nα is equivalent to the Dehn function of a finitely presented group then the first m digits of α are computable in time ≤ C222 Cm for some constant C. This implies that, say, functions nπ+1, ne 2 and nα for all rational numbers α ≥ 4 are equivalent to the Dehn functions of some finitely presented group and that nπ and nα for all rational numbers α ≥ 3 are equivalent to the smallest isodiametric functions of finitely presented groups. Moreover we describe all Dehn functions of finitely presented groups ≻ 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.