Simplicial volume of one-relator groups and stable commutator length

Abstract

A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$, the commutator subgroup of $F(S)$, we introduce the simplicial volume of $\| G_r \|$. We relate this invariant to the stable commutator length $\textrm{scl}_S(r)$ of the element $r \in F(S)$. We show that often (though not always) the linear relationship $\| G_r \| = 4 \cdot \textrm{scl}_S(r) - 2$ holds and that every rational number modulo $1$ is the simplicial volume of a one-relator group. Moreover, we show that this relationship holds approximately for proper powers and for elements satisfying the small cancellation condition $C'(1/N)$, with a multiplicative error of $O(1/N)$. This allows us to prove for random elements of $F(S)'$ of length $n$ that $\| G_r \|$ is $2 \log(2 |S| - 1)/3 \cdot n / \log(n) + o(n/\log(n))$ with high probability, using an analogous result of Calegari-Walker for stable commutator length.