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Abstract:We study the behavior of Dehn functions of finitely presentable groups for presentations with finite generating sets and possibly infinite sets of defining relators. For the free abelian group $\mathbb Z^2$ of rank two on generators $a,b$, we prove that the infinite presentation $\langle a,b \mid [a^{2^k},b],\ k=0,1,2,\ldots\rangle$ has Dehn function of order $n\log n$. We also prove that, for every $0<\alpha<2$, the group $\mathbb Z^2$ admits an infinite presentation on the same two generators whose Dehn function satisfies a global upper bound $\delta(n) \le C n^\alpha + C$ and has matching $n_i^\alpha$-order lower-bound peaks along an infinite sequence of lengths $n_i$. We obtain a similar result, for all $0<\alpha<1$ for torsion-free groups $G$ admitting a finite $C'(1/6)$ small cancellation presentation on the given generators $X$. We also show that the same conclusion holds for an arbitrary finitely generated group $G$ and for some finite generating set $X$ of $G$ and for all $0<\alpha<1$. In particular, these produce continuum many distinct growth types of Dehn functions for presentations of $\mathbb Z^2$ on the standard generators $a,b$.

Submission history

From: Ilya Kapovich [view email]
[v1] Sun, 31 May 2026 15:20:23 UTC (29 KB)
[v2] Wed, 3 Jun 2026 17:03:55 UTC (28 KB)