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Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
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Abstract
Given a group automorphism
ϕ
: Γ → Γ, one has an action of Γ on itself by
ϕ
-twisted conjugacy, namely,
g.x = gxϕ
(
g
-1). The orbits of this action are called
ϕ
-twisted conjugacy classes. One says that Γ has the
R
∞
-property if there are infinitely many
ϕ
-twisted conjugacy classes for every automorphism
ϕ
of Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R
8
-property. Further we show that any (countable) abelian extension of Γ has the R
8
-property where Γ is a torsion free non-elementary hyperbolic group, or SL(
n
; Z); Sp(2
n
; Z) or a principal congruence subgroup of SL(
n
; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.
Title: Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups
Description:
Abstract
Given a group automorphism
ϕ
: Γ → Γ, one has an action of Γ on itself by
ϕ
-twisted conjugacy, namely,
g.
x = gxϕ
(
g
-1).
The orbits of this action are called
ϕ
-twisted conjugacy classes.
One says that Γ has the
R
∞
-property if there are infinitely many
ϕ
-twisted conjugacy classes for every automorphism
ϕ
of Γ.
In this paper we show that SL(n; Z) and its congruence subgroups have the R
8
-property.
Further we show that any (countable) abelian extension of Γ has the R
8
-property where Γ is a torsion free non-elementary hyperbolic group, or SL(
n
; Z); Sp(2
n
; Z) or a principal congruence subgroup of SL(
n
; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.
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