Abstract
The Stein group
F
2
,
3
F_{2,3}
is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form
2
p
3
q
2^{p}3^{q}
(
p
,
q
∈
Z
p,q\in\mathbb{Z}
) and breakpoints in
Z
[
1
6
]
\mathbb{Z}[\frac{1}{6}]
.
This is a natural relative of Thompson’s group 𝐹.
In this paper, we compute the Bieri–Neumann–Strebel–Renz (BNSR) invariants
Σ
m
(
F
2
,
3
)
\Sigma^{m}(F_{2,3})
of the Stein group for all
m
∈
N
m\in\mathbb{N}
.
A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of
F
2
,
3
F_{2,3}
is of type
F
∞
\operatorname{F}_{\infty}
.
Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map
F
2
,
3
→
Z
F_{2,3}\to\mathbb{Z}
is of type
F
∞
\operatorname{F}_{\infty}
, even though there exist maps
F
2
,
3
→
Z
2
F_{2,3}\to\mathbb{Z}^{2}
whose kernels are not even finitely generated.
In terms of BNSR-invariants, this means that every discrete character lies in
Σ
∞
(
F
2
,
3
)
\Sigma^{\infty}(F_{2,3})
, but there exist (non-discrete) characters that do not even lie in
Σ
1
(
F
2
,
3
)
\Sigma^{1}(F_{2,3})
.
To the best of our knowledge,
F
2
,
3
F_{2,3}
is the first group whose BNSR-invariants are known exhibiting these properties.
Complexity among the finitely generated subgroups of Thompson's group
