Irreducible Maps Research Articles

Cubulating hyperbolic free-by-cyclic groups: The irreducible case

Let V be a finite graph, and let ϕ:V→V be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group G. Then G acts freely and cocompactly on a CAT(0) cube complex. Hence, if F is a finite-rank free group and if Φ:F→F is an irreducible monomorphism so that G=F∗Φ is word-hyperbolic, then G acts freely and cocompactly on a CAT(0) cube complex. This holds, in particular, if Φ is an irreducible automorphism with G=F⋊ΦZ word-hyperbolic.

Dynamics on free-by-cyclic groups

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $\epsilon \in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $\Theta_u \subset X$ such that: (1) the inclusion $\Theta_u \to X$ is $\pi_1$-injective and $\pi_1(\Theta_u) = \mathrm{ker}(u)$, (2) $u(\epsilon) = \chi(\Theta_u)$, (3) $\Theta_u \subset X$ is a section of the semiflow and the first return map to $\Theta_u$ is an expanding irreducible train track map representing $\varphi_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{\varphi_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $\varphi_u$ is precisely $\mathfrak H(u)$, (5) if $\varphi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $\varphi_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.

Some Properties of Topological Spaces Related to the Local Density and the Local Weak Density

In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for stratifiable spaces the local density and the local weak density coincide, these cardinal numbers are preserved under open mappings, are inverse invariant of a class of closed irreducible mappings. Moreover, it is showed that the functor of probability measures of finite supports preserves the local density of compacts.

Digraphs and cycle polynomials for free-by-cyclic groups

Let 2 Out.Fn/ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism determines a freeby-cyclic group AD Fn A Z and a homomorphism 2 H 1 .AIZ/. By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, has an open cone neighborhood A in H 1 .AIR/ whose integral points correspond to other fibrations of A whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen’s Teichmuller polynomial that computes the dilatations of all outer automorphisms in A. 57M20

π -Irreducible Mappings and K-Network of Infinite Compacts

In the paper the local density and the local weak density of topological spaces are investigated. It is proved that for a -irreducible mapping f of a topological space X onto a topological space Y the followings hold: d(X) = d(Y ), wd(X) = wd(Y ), ld(X) ld(Y ), lwd(X) lwd(Y ). Moreover, it is showed that the functor of probability measures of nite supports Pn, the functor of the permutation degree SP n G and the functor exp n preserve the cardinality of k-networks of innite compacts.

On Irreducible Maps and Slices

We consider the problem of enumeratingd-irreducible maps,i.e., planar maps all of whose cycles have length at leastd, and such that any cycle of lengthdis the boundary of a face of degreed. We develop two approaches in parallel: the natural approach via substitution, where these maps are obtained from general maps by a replacement of alld-cycles by elementary faces, and a bijective approach via slice decomposition, which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions ofd-irreducible maps with controlled face degrees, summarized in some elegant ‘pointing formula’. We provide an equivalent description ofd-irreducible slices in terms of so-calledd-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees.

A Note on Irreducible Maps with Several Boundaries

We derive a formula for the generating function of $d$-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which is recovered by taking $d=0$. As an application, we obtain an expression for the number of $d$-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges ($d=2$), $4$-irreducible maps and maps of girth at least $6$ ($d=4$). Our derivation is based on a tree interpretation of the various encountered generating functions.

The Auslander bijections: how morphisms are determined by modules

Let be an artin algebra. In his seminal Philadelphia Notes published in 1978, Auslander introduced the concept of morphisms being determined by modules. Auslander was very passionate about these investigations (they also form part of the final chapter of the Auslander–Reiten–Smalo book and could and should be seen as its culmination). The theory presented by Auslander has to be considered as an exciting frame for working with the category of -modules, incorporating all what is known about irreducible maps (the usual Auslander–Reiten theory), but the frame is much wider and allows for example to take into account families of modules—an important feature of module categories. What Auslander has achieved is a clear description of the poset structure of the category of -modules as well as a blueprint for interrelating individual modules and families of modules. Auslander has subsumed his considerations under the heading of “morphisms being determined by modules”. Unfortunately, the wording in itself seems to be somewhat misleading, and the basic definition may look quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander’s powerful results did not gain the attention they deserve. The aim of this survey is to outline the general setting for Auslander’s ideas and to show the wealth of these ideas by exhibiting many examples.

MINIMAL QUASI-F COVERS OF SOME EXTENSION

Observing that every Tychonofi space X has an exten- sion kX which is a weakly Lindelof space and the minimal quasi-F cover QF(kX) of kX is a weakly Lindelof, we show that 'kX : QF(kX) ! kX is a z # -irreducible map and that QF(flX) = flQF(kX). Using these, we prove that QF(kX) = kQF(X) if and only if ' k : kQF(X) ! kX is an onto map and flQF(X) = QF(flX).

BASICALLY DISCONNECTED COVERS OF THE EXTENSION κX OF A SPACE X

Observing that every Tychonoff space X has a weakly Lindel<TEX>$\ddot{o}$</TEX>f extension <TEX>${\kappa}X$</TEX> and the minimal basically diconneted cover <TEX>${\Lambda}{\kappa}X$</TEX> of <TEX>${\kappa}X$</TEX> is weakly Lindel<TEX>$\ddot{o}$</TEX>f, we first show that <TEX>${\Lambda}_{{\kappa}X}:{\Lambda}{\kappa}X{\rightarrow}{\kappa}X$</TEX> is a <TEX>$z^{\sharp}$</TEX>-irreducible map and that <TEX>${\Lambda}{\beta}X={\beta}{\Lambda}{\kappa}X$</TEX>. And we show that <TEX>${\kappa}{\Lambda}X={\Lambda}{\kappa}X$</TEX> if and only if <TEX>${\Lambda}^{\kappa}_X:{\kappa}{\Lambda}X{\rightarrow}{\kappa}X$</TEX> is an onto map and <TEX>${\beta}{\Lambda}X={\Lambda}{\beta}X$</TEX>.

On uniserial modules in the Auslander–Reiten quiver

This article begins the study of irreducible maps involving finite-dimensional uniserial modules over finite-dimensional associative algebras. We work on the classification of irreducible maps between two uniserials over triangular algebras, and give estimates for the number of middle terms of an almost split sequence with a uniserial end term.

Nielsen periodic point theory for periodic maps on orientable surfaces

Let f : F g → F g denote a periodic self map of minimal period m on the orientable surface of genus g with g > 1 . We study the calculation of the Nielsen periodic numbers NP n ( f ) and NΦ n ( f ) . Unlike the general situation of arbitrary maps on such surfaces, strong geometric results of Jiang and Guo allow for straightforward calculations when n ≠ m . However, determining NP m ( f ) involves some surprises. Because f m = id F g , f m has one Nielsen class E m . This class is essential because L ( id F g ) = χ ( F g ) = 2 − 2 g ≠ 0 . If there exists k < m with L ( f k ) ≠ 0 then E m reduces to the essential fixed points of f k . There are maps g (we call them minLef maps) for which L ( g k ) = 0 for all k < m . We show that the period of any minLef map must always divide 2 g − 2 . We prove that for such maps E m reduces algebraically iff it reduces geometrically. This result eliminates one of the most difficult problems in calculating the Nielsen periodic point numbers and gives a complete trichotomy (non-minLef, reducible minLef, and irreducible minLef) for periodic maps on F g . We prove that reducible minLef maps must have even period. For each of the three types of periodic maps we exhibit an example f and calculate both NP n ( f ) and NΦ n ( f ) for all n. The example of an irreducible minLef map is on F 4 and is of maximal period 6. The example of a non-minLef map is on F 2 and has maximal period 12 on F 2 . It is defined geometrically by Wang, and we provide the induced homomorphism and analyze it. The example of an irreducible minLef map is a map of period 6 on F 4 defined by Yang. No algebraic analysis is necessary to prove that this last example is an irreducible minLef map. We explore the algebra involved because it is intriguing in its own right. The examples of reducible minLef maps are simple inversions, which can be applied to any F g . Using these examples we disprove the conjecture from the conclusion of our previous paper.

On the degree of irreducible morphisms

We study the degree of irreducible morphisms in generalized standard convex components of the Auslander–Reiten quiver of an artin algebra with the property that paths with the same origin and end vertices have equal length. We call the components with this last property components with length. In particular, we give two criteria to determine wether the degree of such an irreducible morphism f is finite or infinite. One of them is given in terms of the compositions of f with non-zero maps between modules in the component. The other states that the left degree of an irreducible map f is finite if and only if Ker f belongs to the component. We apply our results to irreducible morphisms over artin algebras of finite representation type and over tame hereditary algebras.

Simplicial maps of graphs that factor through an arc

In this paper it is shown that a simplicial map ϕ from a connected graph into a graph can be factored through an arc if and only if there are a monotone map μ, a sequence π 1, π 2,…, π n of folds, and an irreducible map ψ whose domain is an arc such that ϕ= ψ∘ π n ∘⋯∘ π 2∘ π 1∘ μ. Implicit in this result is a procedure that can be useful for determining whether a simplicial map factors through an arc. For those that do, the procedure produces a factorization. This is demonstrated by means of an example. On the way to the main results, quotient graphs are defined and fundamental results relating them to simplicial maps are proved. Folds are then defined as a specific type of projection onto a quotient graph, generalizing previous definitions.

Irreducible maps and bilinear forms

Bautista showed in 1982 that the possible multiplicities of indecomposable summands of the domains and ranges of irreducible maps between modules over artin algebras are given by numerical invariants of certain bilinear forms associated with the algebra. We obtain further information about these multiplicities by relating the forms to those studied elsewhere in algebra and geometry. One spectacular result is that the allowable multiplicities for some algebras over the field of real numbers depend on J.F. Adams’ determination of the number of linearly independent vector fields on a sphere.

Preprojective Modules and Auslander-Reiten Components

Auslander and Smalø (Auslander, M., Smalø, S. O. (1980). Preprojective modules over artin algebras. J. Algebra 66:61–122) introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. Coelho (Coelho, F. U. (1993). Components of Auslander-Reiten quivers with only preprojective modules. J. Algebra 157: 472–488) studied Auslander-Reiten components containing only hereditarily preprojective modules and gave a number of characterizations of such components. We shall study further these modules by using the description of shapes of semi-stable Auslander-Reitan components; see Liu (Liu, S. (1992). Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc. 45(2):32–54; Liu, S. (1993). Semi-stable components of an Auslander-Reiten quiver. J. London Math. Soc. 47(2):405–416). Our results will imply the result of Coelho (Coelho, F. U. (1993). Components of Auslander-Reiten quivers with only preprojective modules. J. Algebra 157: 472–488, (1.2)) and that of Auslander-Smalø (Auslander, M. and Smalø, S. O. (1980). Preprojective modules over artin algebras, J. Algebra 66:61–122, (9.16)). As an application, moreover, we shall show that a stable Auslander-Reiten component with “few” stable maps in TrD-direction is of shape ℤA ∞.

ON NON-SEMIREGULAR COMPONENTS CONTAINING PATHS FROM INJECTIVE TO PROJECTIVE MODULES

ABSTRACT Let be a component of the Auslander-Reiten quiver of an Artin algebra containing projective and injective modules. Assume that the length of any path from an injective in to a projective in is bounded by some fixed number. We prove here that such component has no oriented cycles and is generalized standard, so containing only finitely many -orbits. Components of the Auslander-Reiten quiver of an Artin algebra containing paths in ind from injective to projective modules have appeared naturally in some classes of algebras such as quasitilted[7] or shod.[4] In both cases, these paths can be refined to paths of irreducible maps and any such path has either none hooks (in the case of a quasitilted algebra) or at most two of them (in the case of a shod algebra). See below for definitions.Here, we are interested in studying the components of such that there exists a number such that any path from an injective to a projective lying on it has at most hooks. We shall see that this is equivalent to the existence of a number such that any path in ind from an injective to a projective lying on it has length at most (Theorem 4.1). Such a component does not have oriented cycles (Corollary 3.4) and it is generalized standard (Theorem 4.3), hence containing only finitely many -orbits.In fact, when considering the existence of oriented cycles in such components, we shall prove the following more general result. If is a component of such that the number of hooks in any path of irreducible maps from an injective in to a projective in is bounded, then has no oriented cycles (Theorem 3.1). This generalizes Liu's main result in,[11] where he shows that any component of such that any path of irreducible maps from an injective in to a projective in is sectional (that is, with no hooks) has no oriented cycles. Observe that Li[8] has also considered such components and has shown that the above condition on the paths from injectives to projectives characterizes the existence of a section in . The proof of Theorem 3.1, however, follows closely some ideas contained in[6], also generalizing results proven there for quasitilted algebras. In a forthcoming paper,[5] we shall make use of the results proven here to study the class of algebras such that the length of any path from an indecomposable injective module to an indecomposable projective module is bounded by some number . This class of algebras contains the quasitilted and the shod algebras. These results were proven in an exchange project between Brazil and Uruguay. Both authors acknowledge financial support given by CNPq and FAPESP, Brazil and PEDECIBA, Uruguay.

Irreducible positive linear maps on operator algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given.

On irreducible mappings into polyhedra

Every mapping f of a normal space X into an arbitrary polyhedron, endowed with the CW-topology, can be approximated by an irreducible mapping g into some subpolyhedron. If f is already irreducible on a subset A of X, one can achieve that f and g coincide on A.

The Königsberg Bridge Problem for Peano Continua

Abstract Peano continua which are images of the unit interval [0,1] or the circle S under a continuous and irreducible map are investigated. Necessary conditions for a space to be the irreducible image of [0,1] are given, and it is conjectured that these conditions are sufficient as well. Also, various results on irreducible images of [0,1] and S are given within some classes of regular curves. Some of them involve inverse limits of inverse sequences of Euler graphs with monotone bonding maps.