CORE
🇺🇦Â
 make metadata, not war
Services
Services overview
Explore all CORE services
Access to raw data
API
Dataset
FastSync
Content discovery
Recommender
Discovery
OAI identifiers
OAI Resolver
Managing content
Dashboard
Bespoke contracts
Consultancy services
Support us
Support us
Membership
Sponsorship
Community governance
Advisory Board
Board of supporters
Research network
About
About us
Our mission
Team
Blog
FAQs
Contact us
research
Orbit decidability and the conjugacy problem for some extensions of groups
Authors
O. Bogopolski
A. Martino
E. Ventura
Publication date
1 October 2007
Publisher
View
on
arXiv
Abstract
Given a short exact sequence of groups with certain conditions,
1
→
F
→
G
→
H
→
1
1\to F\to G\to H\to 1
1
→
F
→
G
→
H
→
1
, we prove that
G
G
G
has solvable conjugacy problem if and only if the corresponding action subgroup
A
⩽
A
u
t
(
F
)
A\leqslant Aut(F)
A
⩽
A
u
t
(
F
)
is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form
Z
2
â‹Š
F
m
\mathbb{Z}^2\rtimes F_m
Z
2
â‹Š
F
m
​
,
F
2
â‹Š
F
m
F_2\rtimes F_m
F
2
​
â‹Š
F
m
​
,
F
n
â‹Š
Z
F_n \rtimes \mathbb{Z}
F
n
​
â‹Š
Z
, and
Z
n
â‹Š
A
F
m
\mathbb{Z}^n \rtimes_A F_m
Z
n
â‹Š
A
​
F
m
​
with virtually solvable action group
A
⩽
G
L
n
(
Z
)
A\leqslant GL_n(\mathbb{Z})
A
⩽
G
L
n
​
(
Z
)
. Also, we give an easy way of constructing groups of the form
Z
4
â‹Š
F
n
\mathbb{Z}^4\rtimes F_n
Z
4
â‹Š
F
n
​
and
F
3
â‹Š
F
n
F_3\rtimes F_n
F
3
​
â‹Š
F
n
​
with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in
A
u
t
(
F
2
)
Aut(F_2)
A
u
t
(
F
2
​
)
is given
Similar works
Full text
Available Versions
RECERCAT
See this paper in CORE
Go to the repository landing page
Download from data provider
oai:recercat.cat:2072/9160
Last time updated on 05/04/2020