Simplicial volume of one-relator groups and stable commutator length

Heuer, Nicolaus; Löh, Clara

doi:10.2140/agt.2022.22.1615

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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

DOI: 10.2140/agt.2022.22.1615

Abstract

A one-relator group is a group $G_{r}$ that admits a presentation $⟨ S ∣ r ⟩$ with a single relation $r$ . One-relator groups form a rich classically studied class of groups in geometric group theory. If $r \in F {(S)}^{'}$ , we introduce a simplicial volume $∥ G_{r} ∥$ for one-relator groups. We relate this invariant to the stable commutator length ${scl}_{S} (r)$ of the element $r \in F (S)$ . We show that often (though not always) the linear relationship $∥ G_{r} ∥ = 4 {scl}_{S} (r) - 2$ 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 $\frac{2}{3} \log (2 | S | - 1) \cdot n ∕ \log n + o (n ∕ \log n)$ with high probability, using an analogous result of Calegari and Walker for stable commutator length.

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Keywords

simplicial volume, one-relator groups, stable commutator, linear programming length

Mathematical Subject Classification 2010

Primary: 20E05, 20F65, 20J05, 57M07, 57M20

References

Publication

Received: 12 November 2019

Revised: 4 December 2020

Accepted: 18 April 2021

Published: 10 October 2022

Authors

Nicolaus Heuer
Department of Mathematics University of Oxford Oxford United Kingdom
https://www.dpmms.cam.ac.uk/~nh441/
Clara Löh
Fakultät für Mathematik Universität Regensburg Regensburg Germany
http://www.mathematik.uni-r.de/loeh

Volume 22, issue 4 (2022)