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# Isoperimetric and Isodiametric Functions of Groups

Mark V. Sapir, Jean-Camille Birget and Eliyahu Rips
Annals of Mathematics
Second Series, Vol. 156, No. 2 (Sep., 2002), pp. 345-466
DOI: 10.2307/3597195
Stable URL: http://www.jstor.org/stable/3597195
Page Count: 122
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## 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 $\leq 2^{2^{Cm}}$ 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 $n^{3\alpha /4}$. On the other hand, if $n^{\alpha}$ is equivalent to the Dehn function of a finitely presented group then α is computable in time $\leq 2^{2^{2^{Cm}}}$ 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 $1/2^{n+1}$ in time at most T(n). This implies that, say, functions $n^{\pi +1}$, $n^{e^{2}}$ and $n^{\alpha}$ for all rational numbers α ≥ 4 are equivalent to Dehn functions of some finitely presented groups and that $n^{\pi}$ 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 $\succ n^{4}$ 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.

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