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Let $G$ be a finitely presented group having Dehn function of type $n^d$ for some natural number $d$. Is it possible to say that $G$ has a presentation the corresponding Dehn function in which is a polynomial of degree $d$?

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    $\begingroup$ What exactly do you mean by "Dehn function of type $n^d$"? Do you mean $O(n^d)$ or $\Theta(n^d)$? Since there are groups with Dehn functions strictly between $n^d$ and $n^{d+1}$ for all $d \ge 2$, it seems feasible that you could have a Dehn function like $\lambda d^3 + \mu d^{5/2} + \cdots$. $\endgroup$
    – Derek Holt
    Nov 15, 2015 at 11:25
  • $\begingroup$ I think the OP's hypothesis means: for some (and hence every) choice of presentation, the Dehn function is bounded above and below by polynomials of degree $d$ and positive leading coefficients. $\endgroup$
    – YCor
    Nov 15, 2015 at 12:17
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    $\begingroup$ ... If a group has a presentation for which Dehn equals $an^3+bn^{5/2}+o(n^{5/2})$, I don't see any reason why it could not have any other presentation for which the Dehn function is exactly a polynomial of degree 3. $\endgroup$
    – YCor
    Nov 15, 2015 at 12:18
  • $\begingroup$ I am interested to know, if there exists a non-trivial example of a f.p group whose Dehn function is explicitly computed. For example, is there a non-trivial known hyperbolic group such that for some finite presentation $\delta(n)=An+B$? $\endgroup$
    – Sh.M1972
    Nov 16, 2015 at 7:00
  • $\begingroup$ What exactly do you mean by non-trivial? Do you regard free groups as trivial? They have Dehn function $0$ (on a free generating set). $\endgroup$
    – Derek Holt
    Nov 16, 2015 at 9:56

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