Isoperimetric and isodiametric functions of groups

TL;DR

This paper explores the deep connections between the asymptotic functions of finitely presented groups and the computational complexity of real numbers, establishing equivalences between Dehn functions and certain computability conditions.

Contribution

It introduces a novel link between Dehn functions of groups and the computability of real number digits, providing characterizations of these functions based on Turing machine complexity.

Findings

Dehn functions equivalent to n^α for computable α with specific time bounds

Smallest isodiametric functions characterized for certain Dehn functions

Classification of Dehn functions greater than n^4 as time functions of Turing machines

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 $\alpha\ge 4$ are computable in time $\le C2^{2^{Cm}}$ for some constant $C>0$ then $n^\alpha$ is equivalent (``big O'') to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is $n^{3/4\alpha}$. On the other hand if $n^\alpha$ is equivalent to the Dehn function of a finitely presented group then the first $m$ digits of $\alpha$ are computable in time $\le C2^{2^{2^{Cm}}}$ for some constant $C$. This implies that, say, functions $n^{\pi+1}$, $n^{e^2}$ and $n^\alpha$ for all rational numbers $\alpha\ge 4$ are equivalent to the Dehn functions of some finitely presented group and that $n^\pi$ and $n^\alpha$ for all rational numbers $\alpha\ge 3$ are equivalent to the smallest isodiametric functions of finitely presented groups. Moreover we describe all Dehn functions of finitely presented groups $\succ n^4$ as time functions of Turing machines modulo two conjectures: \begin{enumerate} \item Every Dehn function is equivalent to a superadditive function. \item The square root of the time function of a Turing machine is equivalent to the time function of a Turing machine. \end{enumerate}