21 research outputs found
Twisted Conjugacy Classes in Lattices in Semisimple Lie Groups
Given a group automorphism , one has an action of
on itself by -twisted conjugacy, namely, .
The orbits of this action are called -conjugacy classes. One says that
has the -property if there are infinitely many
-conjugacy classes for every automorphism of . In this
paper we show that any irreducible lattice in a connected semi simple Lie group
having finite centre and rank at least 2 has the -property.Comment: 6 page
PĂłlyaâCarlson dichotomy for dynamical zeta functions and a twisted BurnsideâFrobenius theorem
For the unitary dual mapping of an automorphism of a torsion-free, finite rank nilpotent group, we prove the PĂłlyaâCarlson dichotomy between rationality and the natural boundary for the analytic behavior of its ArtinâMazur dynamical zeta function. We also establish Gauss congruences for the Reidemeister numbers of the iterations of endomorphisms of groups in this class. Our method is the twisted BurnsideâFrobenius theorem proven in the paper for automorphisms of this class of groups, and a calculation of the Reidemeister numbers via a product formula and profinite completions
Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups
Let Ï : G â G be a group endomorphism where
G is a finitely generated group of exponential growth, and denote
by R(Ï) the number of twisted Ï-conjugacy classes. Felâshtyn and
Hill [7] conjectured that if Ï is injective, then R(Ï) is infinite. This
conjecture is true for automorphisms of non-elementary Gromov
hyperbolic groups, see [17] and [6]. It was showed in [12] that the
conjecture does not hold in general. Nevertheless in this paper,
we show that the conjecture holds for injective homomorphisms for
the family of the Baumslag-Solitar groups B(m,n) where m 6= n
and either m or n is greater than 1, and for automorphisms for the
case m = n > 1. family of the Baumslag-Solitar groups B(m,n)
where m 6= n
The universal functorial equivariant Lefschetz invariant
We introduce the universal functorial equivariant Lefschetz invariant for
endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We
use K_0 of the category of "phi-endomorphisms of finitely generated free
RPi(G,X)-modules". We derive results about fixed points of equivariant
endomorphisms of cocompact proper smooth G-manifolds.Comment: 33 pages; shortened version of the author's PhD thesis, supervised by
Wolfgang Lueck, Westfaelische Wilhelms-Universitaet Muenster, 200
The Conley Conjecture and Beyond
This is (mainly) a survey of recent results on the problem of the existence
of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb
flows. We focus on the Conley conjecture, proved for a broad class of closed
symplectic manifolds, asserting that under some natural conditions on the
manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic
orbits. We discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb flows, the cases
where the conjecture is known to fail, the question of the generic existence of
infinitely many periodic orbits, and local geometrical conditions that force
the existence of infinitely many periodic orbits. We also show how a recently
established variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a low-energy charge
in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur