arXiv research

Nielsen reduction is an algorithm which decomposes any automorphism of a free group into a product of elementary Nielsen transformations. While this may be applied to a mapping class of a surface $S_{g,1}$ with one boundary component, the resulting decomposition in general will not have a topological interpretation. In…

The mapping class group of a genus $g$ surface $Σ_{g,1}$ with one boundary component is known to have a simple yet infinite presentation with generators given by elementary moves called Whitehead moves on so-called marked bordered fatgraphs. In this paper, we introduce an algorithm called "fatgraph Nielsen reduction" w…

Study shows links with trivial Lagrangian are slice, useful for surgery problems.

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen …

Defines Fenchel-Nielsen coordinates for SL(3,C) representations.

Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.

Criterion for realizing groups on Enriques manifolds.

Study strong Nielsen equivalence on punctured disc homeomorphisms.

Approximates call option prices for Barndorff-Nielsen and Shephard model.

For an oriented surface of genus g with b boundary components, we construct a rational map from a subset of C^{6g-6+3b} onto an open algebraic subset of the PSL(2,C)-character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization…

Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.

The paper generalizes Nielsen equivalence to 2-orbifolds.

Extends Alòs' formula to Barndorff-Nielsen and Shephard model.

Solves Nielsen realization problem for hyper-Kähler manifolds.

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 n…

Calculates twist in Teichmüller space using cross ratios.

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.

We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural number N there is a closed hyperbolic 3-manifold whose fundamental group has N d…

Given two maps f1 and f2 from the sphere Sm to an n-manifold N, when are they loose, i.e. when can they be deformed away from one another? We study the geometry of their (generic) coincidence locus and its Nielsen decomposition. On the one hand the resulting bordism class of coincidence data and the corresponding Niels…

Infinitely many unique ways to generate a surface's mapping class group.

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k-1)n to dimension …

We show that the Nielsen number of a map is a knot invariant via representation variety

Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.

The paper proves a theorem about fixed points and relates it to the Nielsen realisation problem.

The theorem connects surface mapping groups to fundamental groupoids.

The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…

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 Cha…

The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.

We introduce Fenchel-Nielsen coordinates on Teicmüller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geom…

Study shows non-spin 4-manifolds where smooth Nielsen realization fails.

For each fixed n>=2 we show how the Nielsen-Thurston classification of mapping classes for a closed surface of genus g>=2 is determined by the sequence of quantum SU(n) representations, when one considers all levels. That this is the case is a consequence of our asymptotic faithfulness property. We here provide explici…

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $\mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel-Nielsen Teichmüller space associated to the pair $(\…

The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.

Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.

Develops methods to simulate option prices for a specific stochastic volatility model.

Solves a problem related to Nielsen realization for certain groups.

Classifies generating systems of Fuchsian groups up to Nielsen equivalence.

The paper defines a new coordinate system for anti-de Sitter structures.

We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a fr…

The paper finds infinitely many 4-manifolds with non-isotopic sections.

Method calculates loop intersections on surfaces.

We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigati…

Study shows Dehn twists on certain 4-manifolds cannot be realized by finite order diffeomorphisms.

Study geodesic flow on symmetric surfaces to determine parabolic type.

The paper provides VIX option pricing and hedging strategies for two stochastic volatility models.

Groups of homotopy equivalences of graphs help realize compact subgroups.

Suppose X,Y are manifolds, f,g:X->Y are maps. The well-known Coincidence Problem studies the coincidence set C={x:f(x)=g(x)}. The number m=dimX-dimY is called the codimension of the problem. More general is the Preimage Problem. For a map f:X->Z and a submanifold Y of Z, it studies the preimage set C={x:f(x) in Y}, and…

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f_1, f_2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC(f_1, f_2) (and MC(f_1, f_2), resp.) of pathcomponents (and of poi…