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. 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 ≤ C2 2 Cm for some constant C > 0 then n α is equivalent ("big O") to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is n 3/4α . On the other hand if n α is equivalent to the Dehn function of a finitely presented group then the first m digits of α are computable in time ≤ C2 2 2 Cm for some constant C. This implies that, say, functions n π+1 , n e 2 and n α for all rational numbers α ≥ 4 are equivalent to the Dehn functions of some finitely presented group and that n π and n α 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 ≻ n 4 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.

Groups with non-simply connected asymptotic cones

“…As an S-machine, G has one state letter k, one tape letter a, and two rules, both of the form [k → ka]. Applying results from [7] to G, we can deduce that G has a cubic isoperimetric function (in fact its Dehn function is exactly n 3 by [5]), and linear isodiametric function. For the sake of completeness, we present below a direct proof of these statements.…”

“…Clearly, u n = 1 in G since k θ n 1 = k θ n 2 = ka n . The corresponding van Kampen diagram has the form of a trapezium [7], [5] with the top and the bottom sides p, p ′ labeled by k n and the left and right sides q, q ′ , labeled by θ n 1 θ −n 2 . The perimeter of that diagram is 6n.…”

“…This group is an S-machine in terminology of [5] or a hub-free interpretation of an S-machine in terminology of [7]. As an S-machine, G has one state letter k, one tape letter a, and two rules, both of the form [k → ka].…”

“…More generally, it is interesting to find out what are the asymptotic cones of S-machines? Topological properties of asymptotic cones may reflect computational properties of the S-machines (for a general definition of an S-machine see [7] or [5]). …”

“…Indeed, it is not even true that polynomial isoperimetric inequality implies linear isodiametric inequality. First examples of groups with polynomial Dehn functions and non-linear isodiametric functions were constructed in [1] and [7] 1 . The question of whether we can guarantee that the asymptotic cones are simply connected by requiring that both the Dehn function is polynomial and the isodiametric function is linear, was open.…”