The paper proves a theorem about fixed points and relates it to the Nielsen realisation problem.
problem The Nielsen realisation problem for aspherical manifolds.
method Proof of a theorem about fixed points and its relation to the Nielsen realisation problem.
result The coincidence of genuine and homotopy fixed points for isometric group actions on nonpositively curved manifolds.
The paper removes a condition for the Nielsen realisation problem using equivariant Poincaré duality.
problem The Nielsen realisation problem for aspherical manifolds.
method Application of equivariant Poincaré duality to cyclic groups of prime order.
result Removal of a technical condition in the Nielsen realisation problem.
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…
We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph Γ, and any finite group G⩽U0(AΓ)⩽Out0(AΓ), we find a non-positively curved cube complex with fundamental group AΓ on which G…
We study dismantling properties of the arc, disc and sphere graphs. We prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out(F_n) fixes a filling (resp. simple) clique in the appropriate graph. We deduce realisation theorems, in particular the Nielsen Real…
Criterion for realizing groups on Enriques manifolds.
problem Realizing groups on Enriques manifolds.
method Using recent developments in Birman-Hilden theory and Nielsen realization for hyper-Kähler manifolds.
result Numerical criterion for realizing groups on Enriques manifolds.
Develops equivariant Seiberg-Witten-Floer cohomology for 3-spheres.
problem Finite group actions on rational homology 3-spheres.
method Equivariant version of Seiberg-Witten-Floer stable homotopy type.
result Definition of d-invariants with Froyshov-type inequality. Solves Nielsen realization problem for hyper-Kähler manifolds.
problem Realization problem for hyper-Kähler manifolds.
method Uses same invariant as for K3 surfaces and determines representation of mapping class group.
result Representation of mapping class group admits a section on its image for some deformation types.
Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.
problem Homotopy coherent Nielsen realization problem for Dehn twists on 4-manifolds
method Using family Seiberg-Witten theory
result Failure of the classical Nielsen realization problem in K3-type 4-manifolds
Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
Solves a problem related to Nielsen realization for certain groups.
problem Whether a cocompact proper topological manifold is equivariantly homotopy equivalent to a classifying space.
method Assumes a zero-dimensional singular set and uses properties of hyperbolic groups and aspherical manifolds.
result Solves the problem for specific groups containing a normal torsion-free subgroup.
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…
Study shows Dehn twists on certain 4-manifolds cannot be realized by finite order diffeomorphisms.
problem Realization of Dehn twists as finite order diffeomorphisms on spin 4-manifolds.
method Use Y. Kato's 10/8-type inequality for involutions and its refinement.
result Dehn twists about specific spheres in certain 4-manifolds are not homotopic to finite order diffeomorphisms.
Solves a problem related to classifying spaces for proper actions and Nielsen Realization.
problem Whether a cocompact proper topological Γ-manifold is equivariantly homotopy equivalent to the classifying space for proper actions.
method Using Poincaré models and assuming a zero-dimensional singular set, the problem is solved in the Poincaré category.
result New results about Brown's problem are obtained under certain conditions on the underlying group.
We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real …
Study mapping class groups of 4-manifolds, proving non-finitely generated and splitting properties.
problem Characterize mapping class groups of 4-manifolds.
method Analyzes intersection lattices, automorphisms, and isotopy classes of diffeomorphisms.
result Proves non-finitely generated and splitting properties of mapping class groups.
Researchers solved a complex problem for a specific type of 4-manifolds.
problem Realizing mapping classes of finite order on del Pezzo surfaces.
method Synthesized results from reflection group theory and 4-manifold topology.
result Provided both positive and negative examples of realizability.
We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two par…
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for mSL(3,C) representations. result Relates to classical and generalized Fenchel-Nielsen coordinates.
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…
Study shows non-spin 4-manifolds where smooth Nielsen realization fails.
problem Existence of non-spin 4-manifolds with non-realizable mapping class groups.
method Investigation of multi-twists, projective twists, and multi-reflections.
result Examples of non-spin 4-manifolds with non-realizable mapping class groups.
Researchers solve Nielsen Realization problems for K3 surfaces in various categories.
problem Realizing finite groups of mapping classes as diffeomorphisms, isometries, or automorphisms in K3 surfaces.
method Introduced a computable invariant LG and constructed an S4 action by isometries. result Some finite groups are realizable while others are not, depending on preserved structures.
Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.
problem Classifying mapping class groups and rational maps.
method Unified proof following Bers' approach.
result Unified proof of Nielsen-Thurston classification.
Study on realizing subgroup twists in 3-manifolds.
problem Realizing subgroups of twist groups in 3-manifolds.
method Analyzing Nielsen realization problem for Twist(M) subgroups, applying to Burnside problem.
result Nontrivial subgroups of Twist(M) are realized by diffeomorphisms if and only if they are cyclic and M is a connected sum of lens spaces.
Study strong Nielsen equivalence on punctured disc homeomorphisms.
problem Characterize strong Nielsen equivalence of periodic points in punctured disc homeomorphisms.
method Braid theory and trace formula application.
result Necessary and sufficient conditions for strong Nielsen equivalence of periodic points.
Approximates call option prices for Barndorff-Nielsen and Shephard model.
problem Calculating exact option prices for complex models is computationally expensive.
method Developed approximate expressions using decomposition formula.
result Approximations are effective as shown by numerical experiments.
Study geodesic flow on symmetric surfaces to determine parabolic type.
problem Determine conditions for a Riemann surface to be of parabolic type.
method Analyze Fenchel-Nielsen coordinates and covering group properties.
result Conditions for a surface to be parabolic are equivalent to specific properties of its Fenchel-Nielsen coordinates.
Enhanced GARCH model uses autoencoder for volatility forecasting.
problem Selecting optimal realised volatility estimator for forecasting.
method Proposes an autoencoder-enhanced Realised GARCH model combining multiple realised measures.
result The model outperforms traditional linear methods in one-step-ahead rolling volatility forecasting.
The paper generalizes Nielsen equivalence to 2-orbifolds.
problem Proving Nielsen equivalence for closed 2-orbifold groups.
method Proving that generating tuples of orbifold fundamental groups are represented by almost orbifold coverings.
result Generalization of Louder's Theorem to closed 2-orbifolds.
Extends Alòs' formula to Barndorff-Nielsen and Shephard model.
problem Modeling call option prices in a stochastic volatility model.
method Uses Alòs' decomposition formula and Ito's formula for an Ornstein-Uhlenbeck model with infinite jumps.
result First Alòs type decomposition formula for Barndorff-Nielsen and Shephard model.
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.
problem Calculating the Fenchel-Nielsen twist in Teichmüller space.
method Using cross ratio coordinates.
result Compact calculation of twist in Teichmüller space.
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∞ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.
The paper solves the Nielsen realization problem for high degree del Pezzo surfaces.
problem Which finite subgroups of the mapping class group of a del Pezzo surface lift to the diffeomorphism group?
method Classification and partial answers for d≥7, equivariant connected sum for d=6. result Complete classification for d≥7, partial answer for d=6. 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…
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 Sg,1 with one boundary component, the resulting decomposition in general will not have a topological interpretation. In…
Infinitely many unique ways to generate a surface's mapping class group.
problem Identifying unique ways to generate mapping class groups.
method Analyzing Nielsen equivalence classes of two-element generators.
result There are 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.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Refined BN-S model improves crude oil hedging with machine learning.
problem Finding optimal hedging strategy for commodity markets.
method Implemented a refined Barndorff-Nielsen and Shephard model with machine learning algorithms.
result The refined model performs better than the classical BN-S model.
The paper studies automorphisms of RAAGs and RACGs, proving properties of their fixed subgroups.
problem Fixed subgroups of automorphisms of RAAGs and RACGs.
method Introducing coarse-median preserving automorphisms and proving properties of fixed subgroups.
result Fixed subgroups of RAAGs and RACGs are finitely generated, undistorted, and quasi-convex.
Innovates rotation index for matrix pairs, solving group action problems.
problem Solving group actions problems, especially Nielsen realization and higher-rank Anosov actions.
method Rotation index and Milnor--Munkres--Novikov pairing applied to Z2 group actions. result Solved specific group action problems using new matrix pair invariant.
The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds M. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest grou…
The theorem connects surface mapping groups to fundamental groupoids.
problem Mapping class groups of bounded surfaces.
method Proving isomorphism between mapping class groups and fundamental groupoid automorphisms.
result Mapping class groups of bounded surfaces are isomorphic to fundamental groupoid automorphisms fixing boundary loops.
Study shows non-cyclic groups of diffeomorphisms can't act on certain 3-manifolds.
problem Realization of finite groups as diffeomorphisms on specific 3-manifolds.
method Analysis of group actions on connected sums of S2imesS1. result No non-cyclic subgroup of twist subgroup can be realized by diffeomorphisms.
The paper shows geometric realisation over specific groups and knots.
problem Geometric realisation of modules over aspherical groups and knots.
method Using extensions of scalars of relation modules and constructing specific 2-complexes.
result Exotic presentations of groups and stably free non-free modules over Baumslag-Solitar groups.