Simplicial volume of one-relator groups and stable commutator length

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 $r\in F(S{)}^{\prime }$, the commutator subgroup of $F(S)$, we introduce the simplicial volume of $\Vert G_r\Vert$. 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 $\Vert G_r\Vert =4\cdot {\text{scl}}_S(r)-2$ 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 $C^{\prime }(1/N)$, with a multiplicative error of $O(1/N)$. This allows us to prove for random elements of $F(S{)}^{\prime }$ of length $n$ that $\Vert G_r\Vert$ 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.

41 pages - comments welcome! v2: Significant changes to v1. The changes include: -A counterexample to the main question of v1. -A theorem about the random simplicial volume of a one-relator group. -A significant restructuring of section 6 (Computational bounds: lallop); v3: to appear in AGT

References in corpus (2)

• The full spectrum of scl on recursively presented groups
• Some remarks on the simplicial volume of nonpositively curved manifolds