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 $0<α<2$, the group $\mathbb Z^2$ admits an infinite presentation on the same two generators whose Dehn function satisfies a global upper bound $δ(n) \le C n^α+ C$ and has matching $n^α$-order lower-bound peaks along an infinite sequence of lengths. For a finite presentation $G=\langle X \mid R\rangle$, let $F(X)$ be the free group on $X$, let $N=\ker(F(X)\to G)$, and let $χ:N\to\mathbb Z$ be a conjugacy-invariant relation invariant with suitable polynomial growth on words in $N$. We prove that then $G$ admits infinite presentations on the same generating set with logarithmic Dehn function in a fine asymptotic sense, and more generally with prescribed polynomial-envelope upper bounds and matching peaks. The general construction gives the stated $\mathbb Z^2$ examples via signed area and gives analogous surface-group examples for every $0<α<1$. In particular, these examples show that, in contrast with the finite-presentation setting, infinite presentations of a fixed finitely generated group on a fixed generating set can exhibit continuum many distinct fine filling-growth behaviors.