Surjective Group Research Articles

Some remarks on algebraic fundamental groups

In this article, we explore several properties of the algebraic fundamental group of a ring and give a surjective group homomorphism from the algebraic fundamental group of the field of real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular rings.

Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications

A toric hyperplane is the preimage of a point in a circle of a continuous surjective group homomorphism from the n-torus to the circle. A toric hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the combinatorial properties of toric hyperplane arrangements on n-tori which are spanning and in general position. Specifically, we describe the symmetry of f-vectors arising in such arrangements and a few applications of the result to count configurations of hyperplanes.

Nonexistence of Surjective Lie Homomorphism Between Complex Special and General Linear Groups

Lie groups are very important objects in algebra, topology and geometry, as it is naturally endowed with two special structures: the algebraic structure of group and the geometric structure of differential manifold. Hence, it is quite meaningful to better know the structure of Lie groups and the actions, and the transformations between them. Two specific theorems about Lie groups and their homomorphism and Lie homomorphism are proved in this paper. First, the fact that there does not exist a surjective Lie group homomorphism between and is introduced. This is done by first having a trial of constructing Lie group surjective homomorphism but fails. Next, this result is further proved by applying topological method that to construct a covering map between and , followed by computing the fundamental group of . Finally, by applying that is a simple group, the fact that there does not exist a surjective group homomorphism between and is demonstrated, together with the statement that is a simple group.

Amenability and acyclicity in bounded cohomology

Johnson’s characterization of amenable groups states that a discrete group \Gamma is amenable if and only if H_b^{n \geq 1}(\Gamma; V) = 0 for all dual normed \mathbb{R}[\Gamma] -modules V . In this paper, we extend the previous result to homomorphisms by proving the converse of the mapping theorem : a surjective group homomorphism \phi \colon \Gamma {\to} K has amenable kernel H if and only if the induced inflation map H^\bullet_b(K; V^H) {\to} H^\bullet_b(\Gamma; V) is an isometric isomorphism for every dual normed \mathbb{R}[\Gamma] -module V . In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms \phi \colon \Gamma \to K with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all Banach \Gamma -modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms, that is, the class of group homomorphisms \phi \colon \Gamma \to K for which the restriction maps in bounded cohomology H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \phi^{-1}V) are isomorphisms for a suitable family of dual normed \mathbb{R}[K] -modules V including the trivial \mathbb{R}[K] -module \mathbb{R} . We then extend the first and third results to topological spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.

C0-transport of flux geometry

The goal of this paper is to study, in a large scale point of view, the flux geometry of a closed symplectic manifold (M,ω): namely, the topological counterpart of the flux homomorphism. Using metrics arising from the decomposition of closed 1-forms with respect to an arbitrary linear section S, we generalize the construction of the group of strong symplectic homeomorphisms. The flux homomorphism for symplectomorphisms is extended to a surjective group homomorphism Sω0 on the group of S-homeomorphisms. We prove that the kernel of Sω0 is path connected, coincides with the subgroup Hameo(M,ω) of all Hamiltonian homeomorphisms and investigate the discreteness of the corresponding flux group SΓω. Later on, without appealing to any lifting map, we give an alternative proof of a result from the classical flux geometry saying that any smooth symplectic isotopy in Ham(M,ω) is a Hamiltonian isotopy. Furthermore under some hypothesis, we prove that any S-topological isotopy in Hameo(M,ω) is a continuous Hamiltonian isotopy. We also proved that any S-topological isotopy with trivial flux is homotopic to a continuous Hamiltonian isotopy, relatively to fixed endpoints.

Quotients of groups of birational transformations of cubic del Pezzo fibrations

We prove that the group of birational transformations of a Del Pezzo fibration of degree 3 over a curve is not simple, by giving a surjective group homomorphism to a free product of infinitely many groups of order 2. As a consequence we also obtain that the Cremona group of rank 3 is not generated by birational maps preserving a rational fibration. Besides, its subgroup generated by all connected algebraic subgroups is a proper normal subgroup.

Residual dimension of nilpotent groups

Abstract The function F G ⁢ ( n ) {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for F N ⁢ ( n ) {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of F N ⁢ ( n ) {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for F N ⁢ ( n ) {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of F N ⁢ ( n ) {\mathrm{F}_{N}(n)} can be fully characterized.

Naturality and definability II

We regard an algebraic construction as a set-theoretically defined map taking structures A to structures B which have A as a distinguished part, in such a way that any isomorphism from A to A' lifts to an isomorphism from B to B'. In general the construction defines B up to isomorphism over A. A construction is uniformisable if the set-theoretic definition can be given in a form such that for each A the corresponding B is determined uniquely. A construction is natural if restriction from B to its part A always determines a map from the automorphism group of B to that of A which is a split surjective group homomorphism. We prove that there is no transitive model of ZFC (Zermelo-Fraenkel set theory with Choice) in which the uniformisable constructions are exactly the natural ones. We construct a transitive model of ZFC in which every uniformisable construction (with a restriction on the parameters in the formulas defining the construction) is ‘weakly’ natural. Corollaries are that the construction of algebraic closures of fields and the construction of divisible hulls of abelian groups have no uniformisations definable in ZFC without parameters.

AUTOMATIC CONTINUITY OF ABSTRACT HOMOMORPHISMS BETWEEN LOCALLY COMPACT AND POLISH GROUPS

We prove results about automatic continuity and openness of abstract surjective group homomorphisms \( K\overset{\varphi }{\to }G, \) where G and K belong to a certain class К of topological groups, and where the kernel of φ satisies a certain topological countability condition. Our results apply in particular to the case where G is a semisimple Lie group or a semisimple compact group, and where К is either the class of all locally compact groups or the class of all Polish groups.

Relative K0 and relative cycle class map

Let F:A→B be an exact functor between small exact categories. We study the zeroth homotopy group K0(F) of the homotopy fiber of the map K(A)→K(B) between K-theory spectra. Under the assumption that F is a cofinal and that B is split exact, we give an explicit description of K0(F) in terms of the triangulated functor Db(A)→Db(B) between the derived categories.We apply it to the pair (X,D) of a scheme X and an affine closed subscheme D of X, and get a description of the relative K0-group K0(X,D) in terms of perfect complexes; it is generated by pairs of two perfect complexes of X together with quasi-isomorphisms along D. This description makes it possible to assign a cycle class in K0(X,D) to a cycle on X not meeting D in an intuitive way. When X is a separated regular scheme of finite type over a field and D is an affine effective Cartier divisor on X, we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus CH⁎(X|D) defined by Binda–Saito to a suitable subquotient of K0(X,D).

Conformal Groups and Vahlen Matrices

This article recalls some facts about the conformal group \({{\rm Conf}(V, Q)}\) of a quadratic space (V, Q); in particular, there is a surjective group morphism \({{\rm O}(V^{\dag}, Q^{\dag}) \rightarrow {\rm Conf}(V, Q)}\), where \({(V^{\dag}, Q^{\dag})}\) is the orthogonal sum of (V, Q) and a hyperbolic plane; its kernel is a group of order two. Then it explains how the elements of \({{\rm O}(V^{\dag}, Q^{\dag})}\) and \({{\rm Conf}(V, Q)}\) can be represented by Vahlen matrices. And finally, it recalls some properties of Vahlen matrices.

Clifford theory and endoisomorphisms

This paper describes the relation between Clifford theory of finite groups over any field, and the Brauer–Clifford group. Let G and G¯ be finite groups, let π:G→G¯ be a surjective group homomorphism with kernel H, and let F be any field. Let K be any extension field of F and let S be an irreducible KH-module. We show that to S is associated in a natural way a specific element [[S,π,F]] of a Brauer–Clifford group defined over π and F. As a tool to prove the existence of this association, and to study its properties, we use endoisomorphisms. These are simply certain isomorphisms of related endomorphism algebras as G¯-algebras over F. We show that two modules for two different finite groups have the same (in an appropriate sense) element of the Brauer–Clifford group if and only if there exists an appropriate endoisomorphism.

On minimal elements for a partial order of prime knots

In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL 2 ( C ) -character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al.

Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic

We show that algebraic analogues of universal group covers, surjective group homomorphisms from a ℚ-vector space to F× with ‘standard kernel’, are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of F and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of ‘Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero’ (Zilber, JLMS 2007), and our proof fills a hole in the proof given there.

Normal Hopf subalgebras in cocycle deformations of finite groups

Let G be a finite group and let π : G → G′ be a surjective group homomorphism. Consider the cocycle deformation L = Hσ of the Hopf algebra H = kG of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra \(k^{G'} \subseteq k^{G}\) corresponds to a Hopf subalgebra \(L' \subseteq L\). Our main result is an explicit necessary and sufficient condition for the normality of L′ in L.

Non-abelian tensor and exterior products modulo q and universal q-central relative extension

The notions of tensor product modulo q and exterior product modulo q of two G-crossed modules, where q is a positive integer and G is a group, are introduced and some properties are given. As an application we give the condition for the existence of a universal q-central relative extension of a surjective group homomorphism and describe this extension as an exterior product modulo q.

Automorphisms of finite fields

Let F be a finite field, and φ: F ∗ → E a surjective group homomorphism from the multiplicative group F ∗ of F to a non-trivial abelian group E. A theorem of McConnel ( Acta Arith. 8 (1963), 127–151) describes the permutations σ of F with the property that φ( σx − σy) = φ( x − y) for all x, y ∈ F, x ≠ y. We give a short proof of this theorem, based on an argument of Bruen and Levinger ( Canad. J. Math. 25 (1973), 1060–1065). In addition, we describe the permutations σ of F for which there exists a permutation κ of E with the property that φ( σx − σy) = κφ( x − y) for all x, y ∈ F, x ≠ y. Finally, we prove a result about automorphisms of the norm form of an arbitrary finite extension of fields.

A measure invariant under group endomorphisms

Let G be a locally compact abelian group. Then there is a finitely additive regular set function m defined on an algebra A of Borel sets in G, m(G) = 1, such that m(T-1F) = m(F) for all F ∈ A and all surjective group endomorphisms T of G onto G.

Projective Limits in Harmonic Analysis

A treatment of induced transformations of measures and measurable functions is presented. Given a diagram $\varphi :G \to H$ in the category of locally compact groups and continuous proper surjective group homomorphisms, functors are produced which on objects are given by $G \to {L^2}(G),{L^1}(G)$, $M(G),W(G)$, denoting, resp., the ${L^2}$-space, ${L^1}$-algebra, measure algebra, and von Neu mann algebra generated by left regular representation of ${L^1}$ on ${L^2}$. All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hilbert spaces and linear transformations of norm $\leqslant 1$ is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of ${W^ \ast }$-algebras and surjective normal $\ast$-algebra homomorphisms.

Homological methods in group varieties

In papers by Stallings [9] and the author [10], [11] the homology theory of groups has been applied to central series, and meaningful group theoretical results have been obtained. In the present paper we show how similar homological methods can be used to obtain interesting results in arbitrary varieties of groups. It is not surprising that the methods are most successful if applied to problems involving central series. It is the nature of a paper introducing new methods to repeat many well known results. However the approach presented here leads to an interestingly unifying point of view. Also, it is possible to simplify the proofs of many well known results. Given a variety ~ , we first define a functor S o from ~ to abelian groups. This functor is defined in terms of the integral second homology group functor H 2 ( , Z). We then prove that a surjective group homomorphism G ~ Q in ~ , with kernel N, gives rise to an exact sequence