Irreducible Maps Research Articles

Infinite metacyclic subgroups of the mapping class group

For g\geq 2 , let \mathrm{Mod}(S_{g}) be the mapping class group of the closed orientable surface S_{g} of genus g . In this paper, we provide necessary and sufficient conditions for a pair of elements in \mathrm{Mod}(S_{g}) to generate an infinite metacyclic subgroup. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of \mathrm{Mod}(S_{g}) with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of \mathrm{Mod}(S_{g}) isomorphic to \mathbb{Z}\rtimes\mathbb{Z}_{m} , \mathbb{Z}_{n}\rtimes\mathbb{Z} , and \mathbb{Z}\rtimes\mathbb{Z} . Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of \mathrm{Mod}(S_{g}) that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class F is either \langle F\rangle or \langle F\rangle\times\langle i\rangle , where i is a hyperelliptic involution.

Riemann surfaces of second kind and effective finiteness theorems

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the finiteness of the number of certain holomorphic objects on closed or punctured Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of second kind is an estimate of the number of irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analog can be interpreted as a quantitatve statement on the limitation for Gromov’s Oka principle. For any finite open Riemann surface X (maybe, of second kind) we give an effective upper bound for the number of irreducible holomorphic mappings up to homotopy from X to the twice punctured complex plane, and an effective upper bound for the number of irreducible holomorphic torus bundles up to isotopy on such a Riemann surface. The bound depends on a conformal invariant of the Riemann surface. If X_{sigma } is the sigma -neighbourhood of a skeleton of an open Riemann surface with finitely generated fundamental group, then the number of irreducible holomorphic mappings up to homotopy from X_{sigma } to the twice punctured complex plane grows exponentially in frac{1}{sigma }.

Irreducible quotient maps from locally compact separable metric spaces

Let X be a Hausdorff quotient of a standard space (that is of a locally compact separable metric space). It is shown that the following are equivalent: (i) X is the image of an irreducible quotient map from a standard space; (ii) X has a sequentially dense subset satisfying two technical conditions involving double sequences; (iii) whenever q:Y→X is a quotient map from a standard space Y, the restriction q⁎|V is an irreducible quotient map from V onto X (where q⁎:Y⁎→X is the pure quotient derived from q, and V is the closure of the set of singleton fibres of Y⁎). The proof uses extensions of the theorems of Whyburn and Zarikian from compact to locally compact standard spaces. The results are new even for quotients of locally compact subsets of the real line.

On Polynomials Counting Essentially Irreducible Maps

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $2b$ and each such cycle of length $2b$ is required to bound a face of degree $2b$. The enumeration of such maps is shown to be again given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in the face degrees with a polynomial dependence on $b$. These polynomials satisfy (generalized) string and dilaton equations, which for $g\leq 1$ uniquely determine them. The proofs rely heavily on a substitution approach by Bouttier and Guitter and the enumeration of planar maps on genus-$g$ surfaces.

Hereditarily irreducible maps

A map f:X→Y from a continuum X onto a continuum Y is said to be hereditarily irreducible, if f(A)⊊f(B) for any subcontinua A and B such that A⊊B. We investigate properties of hereditarily irreducible maps between continua. Special attention is given to maps between graphs and maps from the interval.

On some properties of perfect images of GO-spaces and generalized trees

Let PIGO be the class of perfect images of generalized ordered (GO) spaces. We show that if X∈PIGO, then the following properties are equivalent: X is paracompact; X is metacompact; X is meta-Lindelöf; X is subparacompact; X is submetacompact; X is weak δθ‾-refinable; X is weak θ‾-refinable; X is a D-space; X is an aD-space; X is transitively D; X is linearly D; No closed subspace of X is homeomorphic to a stationary subset of a regular uncountable cardinal; X is submeta-Lindelöf. We also show that if f:X→Y is a perfect mapping such that Y is a semi-stratifiable space and X is a T1-space with a Gδ⁎-diagonal, then X is a semi-stratifiable space.Let PIGT be the class of perfect images of generalized trees with Hausdorff generalized Sorgenfrey topologies. If X∈PIGT and X is a k-semistratifiable (stratifiable) space, then X is metrizable. If X∈PIGT, then X is first-countable if and only if X has a countable pseudocharacter. If f:L→X is a continuous irreducible mapping from a GO-space L (generalized tree L with a Hausdorff generalized Sorgenfrey topology) onto a space X with a regular Gδ-diagonal (Gδ⁎-diagonal), then the space L has a regular Gδ-diagonal (Gδ⁎-diagonal).

On spaces with fragmentable open sets

Recently M.M. Choban et al. introduced spaces with fragmentable open sets and such spaces are denoted by fos-spaces. In this paper the properties of fos-spaces are studied. Among other things we show that a space covered by a locally countable family of closed fos-subspaces is a fos-space. We also prove that every Čech complete fos-space contains a dense completely metrizable space, which improves a result by M.M. Choban et al. Further, it is proved that both continuous closed irreducible mappings and continuous finite-to-one open mappings preserve fos-spaces.

Commuting conjugates of finite-order mapping classes

Let $$\text {Mod}(S_g)$$ be the mapping class group of the closed orientable surface $$S_g$$ of genus $$g\ge 2$$ . In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in $$\text {Mod}(S_g)$$ . As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cyclic cover of $$S_g$$ . Furthermore, we show that any nontrivial torsion element in the centralizer of an irreducible finite order mapping class is of order at most 2. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of $$\text {Mod}(S_g)$$ as isometry groups.

Standard tensor and its applications in problem of singular values of tensors

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.

A new computable sufficient condition for the convergence of subdivision schemes with nonnegative masks

We are interested in nontrivial conditions on the nonnegative masks that guarantee the convergence of the correspondent subdivision schemes. Roughly speaking, a certain convexity of the support of the given mask implies the convergence of the subdivision scheme. Moreover, those conditions are computable. The key of proving our main theorem is to find out an irreducible or primitive mapping on some multi-integer set and to show the uniqueness of this mapping.

Preprojective algebras of tree-type quivers

Let be the path algebra of a tree-type quiver Q, and λ be a nonzero element in a field . We construct irreducible morphisms in the Auslander–Reiten quiver of the transjective component of the bounded derived category of that satisfy what we call the λ-relations. When λ = 1, the relations are known as mesh relations. When , they are known as commutativity relations. We give a new description of the preprojective algebra of and using our technique of constructing irreducible maps together with the results given by Baer–Geigle–Lenzing, Crawley–Boevey, Ringel, and others, we show that for any tree-type quiver, our description is equivalent to several other definitions of preprojective algebras, previously introduced in various contexts.

Correction to: The replacement of irreducible polynomial and affine mapping for the construction of a strong S-box

Correction to: The replacement of irreducible polynomial and affine mapping for the construction of a strong S-box

The replacement of irreducible polynomial and affine mapping for the construction of a strong S-box

Substitution box (S-box) is a critical part of the data encryption and decryption procedures. The primary function of the S-box in advanced encryption standard algorithm is to randomize the 8-bit input into 8-bit output. This paper presents a novel approach to S-box construction based on the replacement of irreducible polynomial and affine mapping. The strength of the created S-box is assessed by applying several standard tests, e.g., balance, bijective, nonlinearity, strict avalanche criterion, and bit independence criterion-nonlinearity. The strength of the S-box outperforms those of available S-boxes.

Endomorphisms, train track maps, and fully irreducible monodromies

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphismof the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application,we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

Iterations of Stone–Čech remainders

Consider a compact space X=L0∪L1 which is a disjoint union of two Lindelöf dense subsets L0, L1, and fix this partition. Then we repeat taking remainders of Stone–Čech compactifications ω1 many times to construct a compact space Ω(X) which is “minimal” w.r.t. the property that Ω(X) admits an irreducible map π onto X such that both π−1(L0) and π−1(L1) are C⁎-embedded in Ω(X). A typical example is the closed interval [0,1]=Q∪P with the partition into the rationals Q and the irrationals P, and we can show that the resultant compact space Ω([0,1]) is not extremally disconnected, hence different from the absolute of [0,1]. We respond to one of van Douwen's questions in [2] that the length of iteration needed for Ω([0,1]) is the first uncountable ordinal ω1.

Irreducible Morphisms Between Modules over a Repetitive Algebras

We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra.

McMullen polynomials and Lipschitz flows for free-by-cyclic groups

Consider a group G and an epimorphism u_0\colon G\to \mathbb Z inducing a splitting of G as a semidirect product ker (u_0)\rtimes_\varphi \mathbb Z with ker (u_0) a finitely generated free group and \varphi\in Out (ker (u_0) ) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized G as \pi_1(X) for an Eilenberg–Maclane 2-complex X equipped with a semiflow \psi , and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant \mathfrak m \in \mathbb Z[H_1(G;\mathbb Z)/ torsion] for (X,\psi) and investigate its properties. Specifically, \mathfrak m determines a convex polyhedral cone \mathcal C_X\subset H^1(G;\mathbb R) , a convex, real-analytic function \mathfrak H\colon \mathcal C_X\to \mathbb R , and specializes to give an integral Laurent polynomial \mathfrak m_u(\zeta) for each integral u\in \mathcal C_X . We show that \mathcal C_X is equal to the "cone of sections" of (X,\psi) (the convex hull of all cohomology classes dual to sections of of \psi ), and that for each (compatible) cross section \Theta_u\subset X with first return map f_u\colon \Theta_u\to \Theta_u , the specialization \mathfrak m_u(\zeta) encodes the characteristic polynomial of the transition matrix of f_u . More generally, for every class u\in \mathcal C_X there exists a geodesic metric d_u and a codimension-1 foliation \Omega_u of X defined by a "closed 1-form" representing u transverse to \psi so that after reparametrizing the flow \psi^u_{s} maps leaves of \Omega_u to leaves via a local e^{s\mathfrak H(u)} -homothety. Among other things, we additionally prove that \mathcal C_X is equal to (the cone over) the component of the BNS-invariant \Sigma(G) containing u_0 and, consequently, that each primitive integral u\in \mathcal C_X induces a splitting of G as an ascending HNN-extension G = Q_u\ast_{\phi_u} with Q_u a finite-rank free group and \phi_u\colon Q_u\to Q_u injective. For any such splitting, we show that the stretch factor of \phi_u is exactly given by e^{\mathfrak H(u)} . In particular, we see that \mathcal C_X and \mathfrak H depend only on the group G and epimorphism u_0 .

Pseudocompleteness in the category of locales

We define pseudocompleteness in the category of locales in a conservative way; so that a space is pseudocomplete in the sense of Oxtoby [24] if and only if the locale it determines is pseudocomplete. We show that a pseudocomplete locale whose Gδ-sublocales are complemented (for instance if it is scattered) is a Baire locale in the sense of Isbell [20]. Our main theorem is that products of pseudocomplete locales are pseudocomplete. Whereas every discrete space is pseudocomplete, and Boolean locales generalize discrete spaces, we demonstrate that not every Boolean locale is pseudocomplete. In [27] Pichardo-Mendoza asks whether pseudocompleteness (in topological spaces) is an invariant of closed irreducible maps. We answer this in the affirmative.

MINIMAL QUASI-F COVERS OF REALCOMPACT SPACES

In this paper, we show that every compactification, which is a quasi-F space, of a space X is a Wallman compactification and that for any compactification K of the space X, the minimal quasi-F cover QFK of K is also a Wallman compactification of the inverse image <TEX>${\Phi}_K^{-1}(X)$</TEX> of the space X under the covering map <TEX>${\Phi}_K:QFK{\rightarrow}K$</TEX>. Using these, we show that for any space X, <TEX>${\beta}QFX=QF{\beta}{\upsilon}X$</TEX> and that a realcompact space X is a projective object in the category <TEX>$Rcomp_{\sharp}$</TEX> of all realcompact spaces and their <TEX>$z^{\sharp}$</TEX>-irreducible maps if and only if X is a quasi-F space.

Connected components of irreducible maps and 1D quantum phases

We investigate elementary topological properties of sets of completely positive (CP) maps that arise in quantum Perron-Frobenius theory. We prove that the set of primitive CP maps of fixed Kraus rank is path-connected and we provide a complete classification of the connected components of irreducible CP maps at given Kraus rank and fixed peripheral spectrum in terms of a multiplicity index. These findings are then applied to analyse 1D quantum phases by studying equivalence classes of translational invariant matrix product states that correspond to the connected components of the respective CP maps. Our results extend the previously obtained picture in that they do not require blocking of physical sites, they lead to analytic paths, and they allow us to decompose into ergodic components and to study the breaking of translational symmetry.