On Dehn functions for infinite group presentations

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 $a,b$.