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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
∈
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
∈
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
2
3 log ( 2 | S | − 1 )
⋅
n ∕ log n
+
o ( n ∕ log n )
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
Publication
Received: 12 November 2019
Revised: 4 December 2020
Accepted: 18 April 2021
Published: 10 October 2022