The R∞ property for pure Artin braid groups Dekimpe, Karel; Lima Gonçalves, Daciberg; Ocampo, Oscar
Monatshefte für Mathematik,
2021/5, Letnik:
195, Številka:
1
Journal Article
Recenzirano
In this paper we prove that all pure Artin braid groups
P
n
(
n
≥
3
) have the
R
∞
property. In order to obtain this result, we analyse the naturally induced morphism
Aut
P
n
⟶
Aut
Γ
2
(
P
n
)
/
Γ
3
...(
P
n
)
which turns out to factor through a representation
ρ
:
S
n
+
1
⟶
Aut
Γ
2
(
P
n
)
/
Γ
3
(
P
n
)
. We can then use representation theory of the symmetric groups to show that any automorphism
α
of
P
n
acts on the free abelian group
Γ
2
(
P
n
)
/
Γ
3
(
P
n
)
via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number
R
(
α
)
of
α
is
∞
.
We study properly discontinuous and cocompact actions of a discrete subgroup \Gamma on a contractible algebraic manifold X on X fixes a point. When the real rank of any simple subgroup of G is at ...most three, we show that \Gamma is virtually polycyclic, we show that the action reduces to an NIL-affine crystallographic action. Specializing to NIL-affine actions, we prove that the generalized Auslander conjecture holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits an NIL-affine crystallographic action.
In this paper we study the Reidemeister spectrum of finitely generated torsion-free 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that ...can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), which describe simply transitive nil-affine actions of Lie groups. We prove existence results for such structures depending on the ...interplay of the algebraic structures of g and n. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we can classify such structures in several cases. We also study commutative structures on low-dimensional Lie algebras and on nilpotent Lie algebras.
Given a group G and an automorphism φ of G, two elements x,y∈G are said to be φ-conjugate if x=gyφ(g)−1 for some g∈G. The number of equivalence classes is the Reidemeister number R(φ) of φ, and if ...R(φ)=∞ for all automorphisms of G, then G is said to have the R∞-property.
A finite simple graph Γ gives rise to the right-angled Artin group AΓ, which has as generators the vertices of Γ and as relations vw=wv if and only if v and w are joined by an edge in Γ. We conjecture that all non-abelian right-angled Artin groups have the R∞-property and prove this conjecture for several subclasses of right-angled Artin groups.
We construct two practical algorithms for twisted conjugacy classes of polycyclic groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given ...endomorphisms, under the condition that their Reidemeister coincidence number is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister coincidence number is finite, or returns “false” if this number is infinite. We also discuss a theoretical extension of these algorithms to polycyclic-by-finite groups.
We study almost inner derivations of 2-step nilpotent Lie algebras of genus 2, i.e., having a 2-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner ...derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not 2 and over the real numbers.
In Dekimpe and Dugardein (J Fixed Point Theory Appl 17:355–370, 2015), Fel’shtyn and Lee (Topol Appl 181:62–103, 2015), the Nielsen zeta function
N
f
(
z
)
has been shown to be rational if
f
is a ...self-map of an infra-solvmanifold of type (R). It is, however, still unknown whether
N
f
(
z
)
is rational for self-maps on solvmanifolds. In this paper, we prove that
N
f
(
z
)
is rational if
f
is a self-map of a (compact) solvmanifold of dimension
≤
5
. In any dimension, we show additionally that
N
f
(
z
)
is rational if
f
is a self-map of an
N
R
-solvmanifold or a solvmanifold with fundamental group of the form
Z
n
⋊
Z
.
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