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Alexander Fel'shtyn

Publications and source records attributed to Alexander Fel'shtyn.

36 records · Page 2Linked to original sources
arXiv2008

Bitwisted Burnside-Frobenius theorem and Dehn conjugacy problem

It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms $ϕ$ and $ψ$ is equal to the number of coincidence points of $\whϕ$ and $\whψ$ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.

math.GR↗
arXiv2008

Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltotic symplectic invariant. A generalisation of the Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed.

math.SG↗
arXiv2008

New directions in Nielsen-Reidemeister theory

The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.

math.GR↗
arXiv2007

Twisted conjugacy classes in Symplectic groups, Mapping class groups and Braid groups(including an Appendix written with Francois Dahmani)

We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $ϕ$ of $G$, where $G$ is one of the above groups, the number of twisted $ϕ$-conjugacy classes is infinite.

math.GR↗
arXiv2007

Twisted conjugacy classes of automorphisms of Baumslag-Solitar groups

Let $ϕ:G \to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(ϕ)$ the number of twisted $ϕ$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that if $ϕ$ is injective, then $R(ϕ)$ is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see \cite{ll} and \cite {fel:1}. It was showed in \cite {gw:2} that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for the Baumslag-Solitar groups $B(m,n)$, where either $|m|$ or $|n|$ is greater than 1 and $|m|\ne |n|$. We also show that in the cases where $|m|=|n|>1$ or $mn=-1$ the conjecture is true for automorphisms. In addition, we derive few results about the coincidence Reidemeister number.

math.GR↗
arXiv2007

Twisted conjugacy classes in R. Thompson's group F

In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers.

math.GR↗
arXiv2006

Twisted Burnside-Frobenius theory for discrete groups

For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^ on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further progress.

math.GR↗
arXiv2006

A twisted Burnside theorem for countable groups and Reidemeister numbers

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Non-commutative Harmonic Analysis. We begin a discussion of the general case (which needs another definition of the dual object). It will be the subject of a forthcoming paper. Some applications and examples are presented.

math.RT↗
arXiv2006

Twisted Burnside theorem for type II_1 groups: an example

The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in a previous paper. In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.

math.RT↗
arXiv2003

Floer Homology, Nielsen Theory and Symplectic Zeta Functions

We describe a connection between symplectic Floer homology for symplectomorphisms of surface and Nielsen fixed point theory. A new zeta functions and asymptotic invariant of symplectic origin are defined. We show that special values of symplectic zeta functions are Reidemeister torsions.

math.SG↗
arXiv2000

Nielsen zeta function, 3-manifolds and asymptotic expansions in Nielsen theory

We prove that the Nielsen zeta function is a rational function or a radical of a rational function for orientation preserving homeomorphisms on closed orientable 3-dimensional manifolds which are special Haken or Seifert manifolds. In the case of pseudo-Anosov homeomorphism of surface we compute an asymptotic for the number of twisted conjugacy classes or for the number of Nielsen fixed point classes whose norm is at most $x$.

math.DS↗
arXiv1998

Reidemeister torsion and Integrable Hamiltonian systems

In this paper we compute the Reidemeister torsion of a isoenergetic surface for the integrable Hamiltonian system on the four-dimensional symplectic manifold. We use the spectral sequence defined by the filtration and following Witten-Floer ideas we bring into play the orbits connecting the critical submanifolds.

math.DS↗
arXiv1998

Trace Formulae, Zeta Functions, Congruences and Reidemeister Torsion in Nielsen Theory

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and torsion free. We continue to study analytical properties of the Nielsen zeta function. We connect the Reidemeister zeta function of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the induced map on the unitary dual of the group. As a consequence we obtain a relation between a special value of the Reidemeister zeta function and a certain Reidemeister torsion. We also prove congruences for Reidemeister numbers of iterates of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group which are the same as those found by Dold for Lefschetz numbers.

math.DG↗
arXiv1996

Dynamical Zeta Functions, Nielsen theory and Reidemeister torsion

The paper consists of four parts. Part I presents a brief survey of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with congruences for the Reidemeister and Nielsen numbers. Part IV deals with the Reidemeister torsion . In Chapter 2 we prove that the Reidemeister zeta function of a group endomorphism is a rational function with functional equation in the following cases: the group is finitely generated and an 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. In Chapter 3 we show that the Nielsen zeta function has a positive radius of convergence which admits a sharp estimate in terms of the topological entropy of the map. For a periodic map of a compact polyhedron we prove that Nielsen zeta function is a radical of a rational function. In sect ion 3.4 and 3.5 we give sufficient conditions under which the Nielsen zeta function coincides with the Reidemeister zeta function and is a rational function with functional equation. In Chapter 5 we prove analog of Dold congruences for Reidemeister and Nielsen numbers. In section 6.2 we establish a connection between the Reidemeister torsion and Reidemeister zeta function. In section 6.3 we establish a connection between the Reidemeister torsion of a mapping torus, the eta-invariant, the Rochlin invariant and the multipliers of the theta function. In section 6.4 we describe with the help of the Reidemeister torsion the connection between the topology of the attraction domain of an attractor and the dynamic of the system on the attractor.

chao-dyn↗
arXiv1995

Trace formulas and dynamical zeta functions in the Nielsen theory

In this paper we prove the trace formulas for the Reidemeister numbers of group endomorphisms in the following cases:the group is finitely generated and an 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 . These results had previously been known only for the finitely generated free abelian group.As a consequence, we obtain under the same conditions on the fundamental group of a compact polyhedron,the trace formulas for the Reidemeister numbers of a continuous map and under suitable conditions the trace formulas for the Nielsen numbers of a continuous map.The trace formula for the Reidemeister numbers implies the rationality of the Reidemeister zeta function. We prove the rationality and functional equation for the Reidemeister zeta function of an endomorphisms of finitely generated torsion free nilpotent group and of a direct sum of a finite group and a finitely generated free abelian group.We give a new proof of the rationality of the Reidemeister zeta function in the case when the group is finitely generated and the endomorphism is eventually commutative and in the case when group is finite.We give also another proof for the positivity of the radius of convergence of the Nielsen zeta function and propose an exast algebraic lower bound estimation for the radius.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 consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion.

chao-dyn↗