Affiliation:
1. Department of Mathematics , State University of Campinas (UNICAMP) , Campinas , SP , Brazil
2. Heinrich-Heine-Universität Düsseldorf (HHU) , Düsseldorf , Germany
Abstract
Abstract
We study the Bieri–Neumann–Strebel–Renz invariants and we prove the following criterion: for groups H and K of type
F
P
n
{FP_{n}}
such that
[
H
,
H
]
⊆
K
⊆
H
{[H,H]\subseteq K\subseteq H}
and a character
χ
:
K
→
ℝ
{\chi:K\to\mathbb{R}}
with
χ
(
[
H
,
H
]
)
=
0
{\chi([H,H])=0}
, we have
[
χ
]
∈
Σ
n
(
K
,
ℤ
)
{[\chi]\in\Sigma^{n}(K,\mathbb{Z})}
if and only if
[
μ
]
∈
Σ
n
(
H
,
ℤ
)
{[\mu]\in\Sigma^{n}(H,\mathbb{Z})}
for every character
μ
:
H
→
ℝ
{\mu:H\to\mathbb{R}}
that extends χ. The same holds for the homotopical invariants
Σ
n
(
-
)
{\Sigma^{n}(-)}
when K and H are groups of type
F
n
{F_{n}}
. We use these criteria to complete the description of the Σ-invariants of the Bieri–Stallings groups
G
m
{G_{m}}
, and more generally to describe the Σ-invariants of the Bestvina–Brady groups. We also show that the “only if” direction of the above criterion holds if we assume only that K is a subnormal subgroup of H, where both groups are of type
F
P
n
{FP_{n}}
. We apply this last result to wreath products.
Funder
Conselho Nacional de Desenvolvimento Científico e Tecnológico
Fundação de Amparo à Pesquisa do Estado de São Paulo
Subject
Applied Mathematics,General Mathematics
Cited by
1 articles.
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