Affiliation:
1. Christ’s College , University of Cambridge , Cambridge , United Kingdom
Abstract
Abstract
We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function
A
Γ
:
N
→
N
\mathcal{A}_{\Gamma}\colon\mathbb{N}\to\mathbb{N}
.
We show that
A
Γ
\mathcal{A}_{\Gamma}
is bounded if and only if Γ has a non-abelian free subgroup.
By contrast, we construct, for any non-decreasing unbounded function
f
:
N
→
N
f\colon\mathbb{N}\to\mathbb{N}
, an elementary amenable lawless group for which
A
Γ
\mathcal{A}_{\Gamma}
grows more slowly than 𝑓.
We produce torsion lawless groups for which
A
Γ
\mathcal{A}_{\Gamma}
is at least linear using Golod–Shafarevich theory and give some upper bounds on
A
Γ
\mathcal{A}_{\Gamma}
for Grigorchuk’s group and Thompson’s group 𝐅.
We note some connections between
A
Γ
\mathcal{A}_{\Gamma}
and quantitative versions of residual finiteness.
Finally, we also describe a function
M
Γ
\mathcal{M}_{\Gamma}
quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups.
By contrast with
A
Γ
\mathcal{A}_{\Gamma}
, there are no groups for which
M
Γ
\mathcal{M}_{\Gamma}
is bounded: we prove a universal lower bound on
M
Γ
(
n
)
\mathcal{M}_{\Gamma}(n)
of the order of
log
(
n
)
\log(n)
.
Funder
European Research Council
Subject
Algebra and Number Theory
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