Simplicial volume of one-relator groups and stable commutator length

A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$withasinglerelation$r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$,thecommutatorsubgroupof$F(S)$, we introduce the simplicial volume of $\mathrm{\parallel }{G}_r\mathrm{\parallel }$. We relate this invariant to the stable commutator length ${\text{scl}}_S(r)$ of the element $r\in F(S)$. We show that often (though not always) the linear relationship $\mathrm{\parallel }{G}_r\mathrm{\parallel }=4\cdot {\text{scl}}_S(r)-2$ 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 $C^{\mathrm{\prime }}(1\mathrm{/}N)$, with a multiplicative error of $O(1\mathrm{/}N)$. This allows us to prove for random elements of $F(S{)}^{\mathrm{\prime }}$ of length $n$ that $\mathrm{\parallel }{G}_r\mathrm{\parallel }$ 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.