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Simplicial volume of one-relator groups and stable commutator length

### Nicolaus Heuer and Clara Löh

Algebraic & Geometric Topology 22 (2022) 1615–1661
##### Abstract

A one-relator group is a group ${G}_{r}$ that admits a presentation $⟨S\mid r⟩$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in geometric group theory. If $r\in F{\left(S\right)}^{\prime }\phantom{\rule{-0.17em}{0ex}}$, we introduce a simplicial volume $\parallel {G}_{r}\parallel$ for one-relator groups. We relate this invariant to the stable commutator length ${\mathrm{scl}}_{S}\left(r\right)$ of the element $r\in F\left(S\right)$. We show that often (though not always) the linear relationship $\parallel {G}_{r}\parallel =4{\mathrm{scl}}_{S}\left(r\right)-2$ 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 ${C}^{\prime }\left(1∕N\right)$, with a multiplicative error of $O\left(1∕N\right)$. This allows us to prove for random elements of $F{\left(S\right)}^{\prime }$ of length $n$ that $\parallel {G}_{r}\parallel$ is $\frac{2}{3}\mathrm{log}\left(2|S|-1\right)\cdot n∕\phantom{\rule{-0.17em}{0ex}}\mathrm{log}n+o\left(n∕\phantom{\rule{-0.17em}{0ex}}\mathrm{log}n\right)$ with high probability, using an analogous result of Calegari and Walker for stable commutator length.

##### Keywords
simplicial volume, one-relator groups, stable commutator, linear programming length
##### Mathematical Subject Classification 2010
Primary: 20E05, 20F65, 20J05, 57M07, 57M20