Imed Kedim - Academia.edu (original) (raw)

Papers by Imed Kedim

Research paper thumbnail of Rigidity of Discontinuous Actions on Diamond Homogeneous Spaces

Let G = R H2n+1 be the diamond group, H a closed Lie subgroup of G and Γ a discontinuous subgroup... more Let G = R H2n+1 be the diamond group, H a closed Lie subgroup of G and Γ a discontinuous subgroup for the homogeneous space G/H. We study in this paper some rigidity properties of the discontinuous action of Γ on G/H. Namely, we show that the strong local rigidity fails to hold on the corresponding parameter space. In addition, when particularly H is dilation-invariant, the local rigidity also fails to hold.

Research paper thumbnail of Stability of Discontinuous Groups Acting on Homogeneous Spaces

Mathematical Notes, 2018

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G... more Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ, G, H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ, H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ, G) and study the case of Heisenberg groups.

Research paper thumbnail of Varietes De Schubert, Varietes Toriques Et Treillis Distributifs

Http Www Theses Fr, 2000

Soit g un groupe algebrique semi-simple sur un corps k algebriquement clos, p un sous-groupe para... more Soit g un groupe algebrique semi-simple sur un corps k algebriquement clos, p un sous-groupe parabolique et w p le quotient du groupe de weyl de g par celui de p. Fixons un poids dominant , associe a p et notons l le fibre en droite g pk g/p. Dans la premiere partie de cette these, pour tout element w de w p, nous construisons une variete projective y w (reunion de varietes toriques) telle que pour tout poids dominant associe a p, il existe un fibre en droite ample sur y w dont le polynome de hilbert est egal a celui de l restreint a la variete de schubert x (w). Dans la seconde partie, nous etudions les proprietes combinatoires de l'ordre de bruhat sur w p, p classique, et, en application, nous construisons une famille plate sur spec (kt) telle que la fibre en (t u), u = 0 est isomorphe a x(w) et la fibre speciale est egale a y w.

Research paper thumbnail of Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation

Mathematical Sciences, 2016

Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces end... more Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations.

Research paper thumbnail of Multivalued fixed point theorem in b-metric spaces and its application to differential inclusions

Filomat, 2018

Our purpose in this paper is to present a fixed point result for multivalued mappings satisfying ... more Our purpose in this paper is to present a fixed point result for multivalued mappings satisfying nonlinear quasi-contractive condition only on related points. Moreover, we provide a qualitative study of well-posedness, limit shadowing property and Ulam-Hyers stability of our fixed point problem. As application, we discuss the existence of a unique solution for a class of differential inclusions.

Research paper thumbnail of Open Problems in Deformation Theory of Discontinuous Groups Acting on Homogeneous Spaces

International Journal of Open Problems in Computer Science and Mathematics, 2013

Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for ... more Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space M = G/H and a discrete subgroup Γ of G isomorphic to Γ, the action of Γ may fail to be properly discontinuous on M (for instance, in the case where H is not compact). To understand this issue, we consider the set R(Γ, G, H) of deformation parameters consisting of all injective homomorphisms of Γ in G, which transform Γ to a discontinuous subgroup of M , so that the related Clifford-Klein forms become manifolds. The group G acts on R(Γ, G, H) by conjugation and the subsequent quotient space T (Γ, G, H) is called the deformation space of the action of Γ on M. The study of these spaces from topological and geometrical points of view, raises many problems of different nature. The main hurdles is to understand the structures of these spaces and some of their topological features. This note aims to record some recent results in the setup of solvable Lie groups and present some open problems in this framework.

Research paper thumbnail of On the local rigidity of discontinuous groups for exponential solvable Lie groups

Advances in Pure and Applied Mathematics, 2013

Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a defo... more Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a deformation of a discontinuous subgroup of G for the homogeneous space G=H may be locally rigid. When G is nilpotent, connected and simply connected, the question whether the Weil-Kobayashi local rigidity fails to hold is posed by Baklouti [Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 9, 173-177]. A positive answer is only provided for some very few cases by now. This note aims to positively answer this question for some new settings. Our study goes even farther to exponential groups. In this case, the local rigidity fails to hold if the automorphism group of the Lie algebra of the syndetic hull of is not solvable. In addition, any deformation of an abelian discontinuous subgroup is shown to be continuously deformable outside the setting of the affine group ax C b.

Research paper thumbnail of The Selberg-Weil-Kobayashi Local Rigidity Theorem for Exponential Lie Groups

International Mathematics Research Notices, 2011

... Unlike the setup of nilpotent Lie groups, we consequently produce examples where the stabilit... more ... Unlike the setup of nilpotent Lie groups, we consequently produce examples where the stability property fails to hold in case of compact Clifford–Klein forms (point 1 in Section 6) and that the properties of rigidity ... 4.1 The terminology of stability in the sense of Kobayashi–Nasrin. ...

Research paper thumbnail of On the Deformation Space of Clifford–Klein Forms of Some Exponential Homogeneous Spaces

International Journal of Mathematics, 2009

Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie gr... more Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie group G. We consider the deformation space [Formula: see text] of a discontinuous subgroup Γ of G for the homogeneous space G/H. When H contains [G, G], we exhibit a description of the space [Formula: see text] which appears to involve GLk(ℝ) as a direct product factor, where k designates the rank of Γ. The moduli space [Formula: see text] is also described. Consequently, we prove in such a setup that the local rigidity property fails to hold globally on [Formula: see text] and that every element of the parameters space is topologically stable.

Research paper thumbnail of Geometric and Harmonic Analysis on Homogeneous Spaces

Lobna Abdelmoula : The Selberg-Weil-Kobayashi local rigidity Theorem for exponential Lie groups. ... more Lobna Abdelmoula : The Selberg-Weil-Kobayashi local rigidity Theorem for exponential Lie groups. A local rigidity Theorem proved by A. Selberg and A. Weil for Riemannian symmetric spaces and generalized by T. Kobayashi for a non-Riemannian homogeneous space G/H, asserts that there are no continuous deformations of a cocompact discontinuous subgroup Γ for G/H in the setting of a linear non-compact semi-simple Lie group G except some few cases : G is not locally isomorphic to SL2(R) for H compact or G is not locally isomorphic to SO(n, 1) or SU(n, 1) for G × G and H = ∆G. When in large contrast G is assumed to be exponential solvable and H ⊂ G a maximal subgroup, we prove an analogue of such a Theorem stating that the local rigidity holds on the parameter space R(Γ, G,H) if and only if G is isomorphic to the two-dimensional group of a ne transformations of the line ax+ b. Remarkably, we do also drop the assumption on Γ to be uniform for G/H. This is a joint work with Ali Baklouti and ...

Research paper thumbnail of A rigidity Theorem and a Stability Theorem for two-step nilpotent Lie groups

Journal of Mathematical Sciences-the University of Tokyo, 2012

Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for ... more Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space � = G/H and any deformation of Γ, the deformed discrete subgroup may fail to be discontinuous for � . To understand this phenomenon in the case when G is a two-step nilpotent Lie group, we provide a stratification of the deformation space of the action of Γ on � , which depends upon the dimensions of G−adjoint orbits. As a direct conse- quence, a rigidity Theorem is given and a certain sufficient condition for the stability property is derived. We also discuss the Hausdorff property of the associated deformation space.

Research paper thumbnail of Fixed point theorems for Maia α-ψ contractive type mappings with applications

We introduce the concept of Maia α-ψ contractive type mappings and establish new fixed point theo... more We introduce the concept of Maia α-ψ contractive type mappings and establish new fixed point theorems. We derive some results for comparable mappings in partially ordered metric spaces, which extend and generalize various known theorems on the topic. As applications, we study the existence and uniqueness of solutions to inhomogeneous Fredholm integral equations of the second kind, and we apply this study to a nonlinear third order two–point boundary value problem.

Research paper thumbnail of Equicontinuity of Maps on Local Dendrites

In this paper, we provide some equivalent conditions of local dendrite maps to be equicontinuous.

Research paper thumbnail of The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

International Journal of Mathematics, 2016

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and gener... more A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that a...

Research paper thumbnail of Eilenberg–Jachymski collections and its first consequences for the fixed point theory

Journal of Fixed Point Theory and Applications

Research paper thumbnail of Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

Open Mathematics

Consider an ordered Banach space and f,g two self-operators defined on the interior of its positi... more Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly \alpha -concave g-monotone or strictly (-\alpha ) -convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

Research paper thumbnail of Some Problems of deformations on three-step nilpotent Lie groups

Hiroshima Mathematical Journal

Research paper thumbnail of Sur le Cortex d'un groupe de Lie nilpotent

Journal of Mathematics of Kyoto University

Research paper thumbnail of A rigidity theorem and a stability theorem for two-step nilpotent Lie groups

Journal of Mathematical Sciences the University of Tokyo, 2012

ABSTRACT For any discontinuous subgroup Γ of a Lie group G, the quotient space Γ∖G/H is said to b... more ABSTRACT For any discontinuous subgroup Γ of a Lie group G, the quotient space Γ∖G/H is said to be a Clifford-Klein form for the homogeneous space G/H· Due to Kobayashi, it is known that any Clifford-Klein form admits a smooth manifold structure for which the quotient map G/H→Γ∖G/H is an open covering and a local diffeomorphism. Let Hom Γ,G be described as follows: Hom Γ,G=φ:Γ→G∣φisahomeomorphism· Moreover, assuming that Hom Γ,G is equipped with the point-wise convergence topology, the parameter space is denoted ℛΓ,G,H which is the set of all φ∈ Hom Γ,G such that φ is injective, φΓ is discrete, φΓ acts properly on G/H and φΓ is fixed point free on G/H· The deformation space of the discontinuous action of Γ on the homogeneous space G/H is the quotient space JΓ,G,H=ℛΓ,G,H/G where the action of G on ℛΓ,G,H is given by g·φ=φ g where φ g γ=gφγg -1 · Let G=expg be a completely solvable Lie group, H=exph a closed connected subgroup of G, Γ a discontinuous subgroup for the homogeneous space G/H and L=expi the smallest connected Lie subgroup of G which contains Γ co-compactly. It is known that the parameter space ℛΓ,G,H is homeomorphic to ℛi,g,h where ℛi,g,h is the set of all injective homomorphisms of the type ψ:i→g such that expψi acts properly on G/H· Moreover, the deformation space JΓ,G,H is also homeomorphic to the quotient space Ji,g,h=ℛi,g,h/G where the action of G on the space ℛΓ,G,H is given by g·ψ=Ad g ∘ψ· In their work, the authors study some geometric and topological properties of the deformation spaces of discontinuous groups acting on some homogeneous spaces in the specific case where G belongs to the class of two-step nilpotent Lie groups. In the first and second section of the paper, the authors provide some background material related to the above discussion. In the third section of the paper, the specific case where G is a two-step nilpotent Lie group, H a connected subgroup of G and Γ a discontinuous subgroup of G/H is addressed. Using a natural stratified structure of the Lie algebra, they decompose g as follows: g=𝔷⊕g ' where𝔷= cent g· The above decomposition is used by the authors to first obtain a precise description of the parameter space ℛi,g,h· In fact they show that it is the disjoint union of two sets ℛ 1 and ℛ 2 where an explicit description of the sets is given. Secondly, they provide a precise description of the deformation space Ji,g,h which is explicitly given as a union of semi-algebraic sets. As a result, the following conjecture is confirmed in the fourth section of the paper in the particular case of two-step nilpotent Lie groups. Let G be a connected simply connected nilpotent Lie group, H a connected subgroup and Γ a non-trivial discontinuous subgroup for G/H· Then the local rigidity fails. We recall that for φ∈ℛΓ,G,H, the discontinuous subgroup φΓ is said to be locally rigid as a discontinuous group of G/H if the orbit of φ through the inner conjugation is open in ℛΓ,G,H· Also, φ is said to be topologically stable if there is an open set in Hom Γ,G which contains φ and is also contained in ℛΓ,G,H· In the fifth section, the authors study the stability property in the restricted case of two-step nilpotent Lie groups. Finally, in the last section of the article, the authors show that the deformation Ji,g,h is in fact a Hausdorff space. Throughout the paper, several examples are computed very explicitly.

Research paper thumbnail of On the cortex of a nilpotent Lie group

Journal of Mathematics Kyoto University, 2009

Research paper thumbnail of Rigidity of Discontinuous Actions on Diamond Homogeneous Spaces

Let G = R H2n+1 be the diamond group, H a closed Lie subgroup of G and Γ a discontinuous subgroup... more Let G = R H2n+1 be the diamond group, H a closed Lie subgroup of G and Γ a discontinuous subgroup for the homogeneous space G/H. We study in this paper some rigidity properties of the discontinuous action of Γ on G/H. Namely, we show that the strong local rigidity fails to hold on the corresponding parameter space. In addition, when particularly H is dilation-invariant, the local rigidity also fails to hold.

Research paper thumbnail of Stability of Discontinuous Groups Acting on Homogeneous Spaces

Mathematical Notes, 2018

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G... more Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ, G, H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ, H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ, G) and study the case of Heisenberg groups.

Research paper thumbnail of Varietes De Schubert, Varietes Toriques Et Treillis Distributifs

Http Www Theses Fr, 2000

Soit g un groupe algebrique semi-simple sur un corps k algebriquement clos, p un sous-groupe para... more Soit g un groupe algebrique semi-simple sur un corps k algebriquement clos, p un sous-groupe parabolique et w p le quotient du groupe de weyl de g par celui de p. Fixons un poids dominant , associe a p et notons l le fibre en droite g pk g/p. Dans la premiere partie de cette these, pour tout element w de w p, nous construisons une variete projective y w (reunion de varietes toriques) telle que pour tout poids dominant associe a p, il existe un fibre en droite ample sur y w dont le polynome de hilbert est egal a celui de l restreint a la variete de schubert x (w). Dans la seconde partie, nous etudions les proprietes combinatoires de l'ordre de bruhat sur w p, p classique, et, en application, nous construisons une famille plate sur spec (kt) telle que la fibre en (t u), u = 0 est isomorphe a x(w) et la fibre speciale est egale a y w.

Research paper thumbnail of Coincidence and common fixed point results for β-quasi contractive mappings on metric spaces endowed with binary relation

Mathematical Sciences, 2016

Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces end... more Coincidence and common fixed point theorems for b-quasi contractive mappings on metric spaces endowed with binary relations and involving suitable comparison functions are presented. Our results generalize, improve, and extend several recent results. As an application, we study the existence of solutions for some class of integral equations.

Research paper thumbnail of Multivalued fixed point theorem in b-metric spaces and its application to differential inclusions

Filomat, 2018

Our purpose in this paper is to present a fixed point result for multivalued mappings satisfying ... more Our purpose in this paper is to present a fixed point result for multivalued mappings satisfying nonlinear quasi-contractive condition only on related points. Moreover, we provide a qualitative study of well-posedness, limit shadowing property and Ulam-Hyers stability of our fixed point problem. As application, we discuss the existence of a unique solution for a class of differential inclusions.

Research paper thumbnail of Open Problems in Deformation Theory of Discontinuous Groups Acting on Homogeneous Spaces

International Journal of Open Problems in Computer Science and Mathematics, 2013

Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for ... more Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space M = G/H and a discrete subgroup Γ of G isomorphic to Γ, the action of Γ may fail to be properly discontinuous on M (for instance, in the case where H is not compact). To understand this issue, we consider the set R(Γ, G, H) of deformation parameters consisting of all injective homomorphisms of Γ in G, which transform Γ to a discontinuous subgroup of M , so that the related Clifford-Klein forms become manifolds. The group G acts on R(Γ, G, H) by conjugation and the subsequent quotient space T (Γ, G, H) is called the deformation space of the action of Γ on M. The study of these spaces from topological and geometrical points of view, raises many problems of different nature. The main hurdles is to understand the structures of these spaces and some of their topological features. This note aims to record some recent results in the setup of solvable Lie groups and present some open problems in this framework.

Research paper thumbnail of On the local rigidity of discontinuous groups for exponential solvable Lie groups

Advances in Pure and Applied Mathematics, 2013

Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a defo... more Let H be an arbitrary closed connected subgroup of an exponential solvable Lie group. Then a deformation of a discontinuous subgroup of G for the homogeneous space G=H may be locally rigid. When G is nilpotent, connected and simply connected, the question whether the Weil-Kobayashi local rigidity fails to hold is posed by Baklouti [Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 9, 173-177]. A positive answer is only provided for some very few cases by now. This note aims to positively answer this question for some new settings. Our study goes even farther to exponential groups. In this case, the local rigidity fails to hold if the automorphism group of the Lie algebra of the syndetic hull of is not solvable. In addition, any deformation of an abelian discontinuous subgroup is shown to be continuously deformable outside the setting of the affine group ax C b.

Research paper thumbnail of The Selberg-Weil-Kobayashi Local Rigidity Theorem for Exponential Lie Groups

International Mathematics Research Notices, 2011

... Unlike the setup of nilpotent Lie groups, we consequently produce examples where the stabilit... more ... Unlike the setup of nilpotent Lie groups, we consequently produce examples where the stability property fails to hold in case of compact Clifford–Klein forms (point 1 in Section 6) and that the properties of rigidity ... 4.1 The terminology of stability in the sense of Kobayashi–Nasrin. ...

Research paper thumbnail of On the Deformation Space of Clifford–Klein Forms of Some Exponential Homogeneous Spaces

International Journal of Mathematics, 2009

Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie gr... more Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie group G. We consider the deformation space [Formula: see text] of a discontinuous subgroup Γ of G for the homogeneous space G/H. When H contains [G, G], we exhibit a description of the space [Formula: see text] which appears to involve GLk(ℝ) as a direct product factor, where k designates the rank of Γ. The moduli space [Formula: see text] is also described. Consequently, we prove in such a setup that the local rigidity property fails to hold globally on [Formula: see text] and that every element of the parameters space is topologically stable.

Research paper thumbnail of Geometric and Harmonic Analysis on Homogeneous Spaces

Lobna Abdelmoula : The Selberg-Weil-Kobayashi local rigidity Theorem for exponential Lie groups. ... more Lobna Abdelmoula : The Selberg-Weil-Kobayashi local rigidity Theorem for exponential Lie groups. A local rigidity Theorem proved by A. Selberg and A. Weil for Riemannian symmetric spaces and generalized by T. Kobayashi for a non-Riemannian homogeneous space G/H, asserts that there are no continuous deformations of a cocompact discontinuous subgroup Γ for G/H in the setting of a linear non-compact semi-simple Lie group G except some few cases : G is not locally isomorphic to SL2(R) for H compact or G is not locally isomorphic to SO(n, 1) or SU(n, 1) for G × G and H = ∆G. When in large contrast G is assumed to be exponential solvable and H ⊂ G a maximal subgroup, we prove an analogue of such a Theorem stating that the local rigidity holds on the parameter space R(Γ, G,H) if and only if G is isomorphic to the two-dimensional group of a ne transformations of the line ax+ b. Remarkably, we do also drop the assumption on Γ to be uniform for G/H. This is a joint work with Ali Baklouti and ...

Research paper thumbnail of A rigidity Theorem and a Stability Theorem for two-step nilpotent Lie groups

Journal of Mathematical Sciences-the University of Tokyo, 2012

Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for ... more Let G be a Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup Γ for the homogeneous space � = G/H and any deformation of Γ, the deformed discrete subgroup may fail to be discontinuous for � . To understand this phenomenon in the case when G is a two-step nilpotent Lie group, we provide a stratification of the deformation space of the action of Γ on � , which depends upon the dimensions of G−adjoint orbits. As a direct conse- quence, a rigidity Theorem is given and a certain sufficient condition for the stability property is derived. We also discuss the Hausdorff property of the associated deformation space.

Research paper thumbnail of Fixed point theorems for Maia α-ψ contractive type mappings with applications

We introduce the concept of Maia α-ψ contractive type mappings and establish new fixed point theo... more We introduce the concept of Maia α-ψ contractive type mappings and establish new fixed point theorems. We derive some results for comparable mappings in partially ordered metric spaces, which extend and generalize various known theorems on the topic. As applications, we study the existence and uniqueness of solutions to inhomogeneous Fredholm integral equations of the second kind, and we apply this study to a nonlinear third order two–point boundary value problem.

Research paper thumbnail of Equicontinuity of Maps on Local Dendrites

In this paper, we provide some equivalent conditions of local dendrite maps to be equicontinuous.

Research paper thumbnail of The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

International Journal of Mathematics, 2016

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and gener... more A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that a...

Research paper thumbnail of Eilenberg–Jachymski collections and its first consequences for the fixed point theory

Journal of Fixed Point Theory and Applications

Research paper thumbnail of Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

Open Mathematics

Consider an ordered Banach space and f,g two self-operators defined on the interior of its positi... more Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly \alpha -concave g-monotone or strictly (-\alpha ) -convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

Research paper thumbnail of Some Problems of deformations on three-step nilpotent Lie groups

Hiroshima Mathematical Journal

Research paper thumbnail of Sur le Cortex d'un groupe de Lie nilpotent

Journal of Mathematics of Kyoto University

Research paper thumbnail of A rigidity theorem and a stability theorem for two-step nilpotent Lie groups

Journal of Mathematical Sciences the University of Tokyo, 2012

ABSTRACT For any discontinuous subgroup Γ of a Lie group G, the quotient space Γ∖G/H is said to b... more ABSTRACT For any discontinuous subgroup Γ of a Lie group G, the quotient space Γ∖G/H is said to be a Clifford-Klein form for the homogeneous space G/H· Due to Kobayashi, it is known that any Clifford-Klein form admits a smooth manifold structure for which the quotient map G/H→Γ∖G/H is an open covering and a local diffeomorphism. Let Hom Γ,G be described as follows: Hom Γ,G=φ:Γ→G∣φisahomeomorphism· Moreover, assuming that Hom Γ,G is equipped with the point-wise convergence topology, the parameter space is denoted ℛΓ,G,H which is the set of all φ∈ Hom Γ,G such that φ is injective, φΓ is discrete, φΓ acts properly on G/H and φΓ is fixed point free on G/H· The deformation space of the discontinuous action of Γ on the homogeneous space G/H is the quotient space JΓ,G,H=ℛΓ,G,H/G where the action of G on ℛΓ,G,H is given by g·φ=φ g where φ g γ=gφγg -1 · Let G=expg be a completely solvable Lie group, H=exph a closed connected subgroup of G, Γ a discontinuous subgroup for the homogeneous space G/H and L=expi the smallest connected Lie subgroup of G which contains Γ co-compactly. It is known that the parameter space ℛΓ,G,H is homeomorphic to ℛi,g,h where ℛi,g,h is the set of all injective homomorphisms of the type ψ:i→g such that expψi acts properly on G/H· Moreover, the deformation space JΓ,G,H is also homeomorphic to the quotient space Ji,g,h=ℛi,g,h/G where the action of G on the space ℛΓ,G,H is given by g·ψ=Ad g ∘ψ· In their work, the authors study some geometric and topological properties of the deformation spaces of discontinuous groups acting on some homogeneous spaces in the specific case where G belongs to the class of two-step nilpotent Lie groups. In the first and second section of the paper, the authors provide some background material related to the above discussion. In the third section of the paper, the specific case where G is a two-step nilpotent Lie group, H a connected subgroup of G and Γ a discontinuous subgroup of G/H is addressed. Using a natural stratified structure of the Lie algebra, they decompose g as follows: g=𝔷⊕g ' where𝔷= cent g· The above decomposition is used by the authors to first obtain a precise description of the parameter space ℛi,g,h· In fact they show that it is the disjoint union of two sets ℛ 1 and ℛ 2 where an explicit description of the sets is given. Secondly, they provide a precise description of the deformation space Ji,g,h which is explicitly given as a union of semi-algebraic sets. As a result, the following conjecture is confirmed in the fourth section of the paper in the particular case of two-step nilpotent Lie groups. Let G be a connected simply connected nilpotent Lie group, H a connected subgroup and Γ a non-trivial discontinuous subgroup for G/H· Then the local rigidity fails. We recall that for φ∈ℛΓ,G,H, the discontinuous subgroup φΓ is said to be locally rigid as a discontinuous group of G/H if the orbit of φ through the inner conjugation is open in ℛΓ,G,H· Also, φ is said to be topologically stable if there is an open set in Hom Γ,G which contains φ and is also contained in ℛΓ,G,H· In the fifth section, the authors study the stability property in the restricted case of two-step nilpotent Lie groups. Finally, in the last section of the article, the authors show that the deformation Ji,g,h is in fact a Hausdorff space. Throughout the paper, several examples are computed very explicitly.

Research paper thumbnail of On the cortex of a nilpotent Lie group

Journal of Mathematics Kyoto University, 2009