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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

A one-relator group is a group Gr that admits a presentation Sr with a single relation r. One-relator groups form a rich classically studied class of groups in geometric group theory. If r F(S), we introduce a simplicial volume Gr for one-relator groups. We relate this invariant to the stable commutator length scl S(r) of the element r F(S). We show that often (though not always) the linear relationship Gr = 4scl 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(1N), with a multiplicative error of O(1N). This allows us to prove for random elements of F(S) of length n that Gr is 2 3 log (2|S| 1) nlog n + o(nlog n) with high probability, using an analogous result of Calegari and Walker for stable commutator length.

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simplicial volume, one-relator groups, stable commutator, linear programming length
Mathematical Subject Classification 2010
Primary: 20E05, 20F65, 20J05, 57M07, 57M20
Received: 12 November 2019
Revised: 4 December 2020
Accepted: 18 April 2021
Published: 10 October 2022
Nicolaus Heuer
Department of Mathematics
University of Oxford
United Kingdom
Clara Löh
Fakultät für Mathematik
Universität Regensburg