Simplicial volume of one-relator groups and stable commutator length

TL;DR

This paper studies the simplicial volume of one-relator groups, relating it to stable commutator length, and demonstrates that their relationship often follows a linear pattern with concrete probabilistic and approximation results.

Contribution

It introduces the simplicial volume for one-relator groups, establishes its relationship with stable commutator length, and analyzes its behavior for random elements and under small cancellation conditions.

Findings

Linear relationship between simplicial volume and stable commutator length often holds.

Every rational number modulo 1 can be realized as a simplicial volume.

Simplicial volume of random elements grows proportionally to n / log(n).

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.