The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion

Fel′shtyn, Alexander; Hill, Richard M.

doi:10.1007/bf00961408

Cited by 78 publications

(76 citation statements)

References 19 publications

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“…In 1994, A. Fel'shtyn and R. Hill [9] conjectured that for a finitely generated group π of exponential growth, if ϕ : π → π is injective then R(ϕ) = ∞. Using techniques from geometric group theory, G. Levitt and M. Lustig [24] showed that if π is finitely generated torsion-free non-elementary Gromov hyperbolic then every automorphism ϕ ∈ Aut(π) must have R(ϕ) = ∞.…”

Section: Introductionmentioning

confidence: 99%

Twisted conjugacy classes in nilpotent groups

Gonçalves¹,

Wong²

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. A group is said to have the R ∞ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R ∞ property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n ≥ 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the R ∞ property.The R ∞ property for virtually abelian and for C-nilpotent groups are also discussed.

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Section: Introductionmentioning

confidence: 99%

Twisted conjugacy classes in nilpotent groups

Gonçalves¹,

Wong²

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…periodic point classes of f . This was previously known for the finitely generated Abelian groups and finite groups [17].…”

mentioning

confidence: 62%

“…In the paper [17] we have connected the Reidemeister number of an endomorphism φ with the Lefschetz number of the dual map. From this we have the following trace formula:…”

Section: Theoremmentioning

confidence: 99%

Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion

Fel′shtyn

Hill

1999

Banach Center Publ.

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Abstract. In this paper we prove trace formulas for the Reidemeister numbers of group endomorphisms and the rationality of the Reidemeister zeta function in the following cases: the group is finitely generated and the endomorphism is eventually commutative; the group is finite; the group is a direct sum of a finite group and a finitely generated free Abelian group; the group is finitely generated, nilpotent and torsion free. We connect the Reidemeister zeta function of an endomorphism of a direct sum of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the unitary dual map, and as a consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion. We also prove congruences for Reidemeister numbers which are the same as those found by Dold for Lefschetz numbers.

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“…assigned to a mapping class g via zeta function F φ (t) of a monotone (or weakly monotone) representative φ ∈ Symp m (M, ω) of g. Symplectomorphisms φ n are also monotone (weakly monotone) for all n > 0 (see [19,4]) so, symplectic zeta function F φ (t) is an invariant as φ is deformed through monotone (or weakly monotone) symplectomorphisms in g. These imply that we have a symplectic Floer homology invariant F g (t) canonically assigned to each mapping class g. A motivation for the definition of this zeta function was a connection [11,19] between Nielsen numbers and Floer homology and nice analytic properties of Nielsen zeta function [33,8,11,9,10,13,14].…”

Section: Radius Of Convergence Of the Symplectic Zeta Functionmentioning

confidence: 99%

The growth rate of symplectic Floer homology

Fel′shtyn

2012

J. Fixed Point Theory Appl.

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Abstract. The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudoAnosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. This also establishes a connection between Floer homology and geometry of 3-manifolds.Mathematics Subject Classification. 37C25, 53D, 37C30, 55M20.

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The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion

Cited by 78 publications

References 19 publications

Twisted conjugacy classes in nilpotent groups

Twisted conjugacy classes in nilpotent groups

Dynamical zeta functions, congruences in Nielsen theory and Reidemeister torsion

The growth rate of symplectic Floer homology

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