Finite Index Research Articles

Unleashing the potential of water-insoluble Cu2+-crosslinked chitosan nanocomposite film for enhanced antibacterial and flame-retardant properties

Chitosan (CS) has received widespread attention in various fields due to its excellent antimicrobial properties, inherent biodegradability and biocompatibility. However, the water swelling property of chitosan films reduces their performance and becomes a major obstacle for their practical application. Herein, we develop a water-insoluble montmorillonite/chitosan nanocomposite film (CS/MMT-Cu) with anti-bacterial and fire-retardant properties via Cu2+-crosslinking technique. Cu2+ ions are coordinated with amino and hydroxyl groups to crosslink the chitosan chains and bind to the montmorillonite (MMT) via electrostatic forces. As a result, the content of MMT has a significant effect on the mechanical and flame-retardant properties of CS/MMT-Cu films, in which the maximum tensile strength and finite oxygen index are 30.6 MPa and 40.5 %, respectively. In addition, the CS/MMT-Cu films exhibit good anti-bacterial properties. Overall, the prepared CS/MMT-Cu films have a broad application prospect in various fields, such as flame-retardant material, packaging and coatings.

Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index

We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral n-varifolds V in Riemannian manifolds which have both: (i) finite index on their smoothly embedded part; and (ii) Hn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^{n-1}$$\\end{document}-null singular set. A direct consequence of such a bound is a bound on the upper Minkowski content of the singular set of such a varifold in terms of its total mass and its index. Such a result improves on known bounds, namely the corresponding bound on the Hn-7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^{n-7}$$\\end{document}-measure of the singular set established by Song (Duke Math J 172(11):2073–2147, 2023), as well as the same bounds established by Naber and Valtorta (J Eur Math Soc 22(10):3305–3382, 2020) for codimension one area minimising currents. Our results also provide more structural information on the singular set for codimension one integral varifolds with finite index (on the regular part) and no classical singularities established by Wickramasekera (Ann Math 179:843–1007, 2014).

[formula omitted]-shadows for the gentle version [formula omitted] of the Grothendieck-Teichmueller group

Let B3 be the Artin braid group on 3 strands and PB3 be the corresponding pure braid group. In this paper, we construct the groupoid GTSh of GT-shadows for a (possibly more tractable) version GTˆ0 of the Grothendieck-Teichmueller group GTˆ introduced in paper [12] by D. Harbater and L. Schneps. We call this group the gentle version of GTˆ and denote it by GTˆgen. The objects of GTSh are finite index normal subgroups N of B3 satisfying the condition N≤PB3. Morphisms of GTSh are called GT-shadows and they may be thought of as approximations to elements of GTˆgen. We show how GT-shadows can be obtained from elements of GTˆgen and prove that GTˆgen is isomorphic to the limit of a certain functor defined in terms of the groupoid GTSh. Using this result, we get a criterion for identifying genuine GT-shadows.

Centralisers of linear growth automorphisms of free groups

Abstract In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

Minimal compact group actions on C∗-algebras with simple fixed point algebras

The notion of quasi-product actions of a compact group on a C[Formula: see text]-algebra was introduced by Bratteli et al. in their attempt to seek an equivariant analogue of Glimm’s characterization of non-type I C[Formula: see text]-algebras. We show that a faithful minimal action of a second countable compact group on a separable C[Formula: see text]-algebra is quasi-product whenever its fixed point algebra is simple. This was previously known only for compact abelian groups and for profinite groups. Our proof relies on a subfactor technique applied to finite index inclusions of simple C[Formula: see text]-algebras in the purely infinite case, and also uses ergodic actions of compact groups in the general case. As an application, we show that if moreover the fixed point algebra is a Kirchberg algebra, such an action is always isometrically shift-absorbing, and hence is classifiable by the equivariant KK-theory due to a recent result of Gabe-Szabó.

Cyclic‐Schottky strata of Schottky space

AbstractSchottky space , where is an integer, is a connected complex orbifold of dimension ; it provides a parametrization of the ‐conjugacy classes of Schottky groups of rank . The branch locus , consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If , then there is a Kleinian group containing as a normal subgroup of index some prime integer . The structural description, in terms of Klein–Maskit Combination Theorems, of such a group is completely determined by a triple , where are integers such that . For each such tuple , there is a corresponding cyclic‐Schottky stratum . It is known that is connected. In this paper, for , we study the connectivity of these .

The action of [formula omitted]-shadows on child's drawings

GT-shadows [8] are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group GTˆ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB4, that are normal in B4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of GTˆ. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, GTˆ, and the absolute Galois group GQ of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid GTSh♡ of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over Q. Finally, we describe selected examples of non-Abelian child's drawings.

Classification of solutions to some elliptic equations with Robin boundary condition and finite Morse index

We deal here with the non-existence of weak finite Morse index solutions to the following nonlinear elliptic equation −Δu=(1+|x|α)|u|p−1uinRN, or in half space with Robin boundary condition, where N≥2, α>−2 and p>1.

ADJUNCTION OF ROOTS TO PROFINITE GROUPS

In this paper, when $G'$ is a group obtained by adjoining a $n$th-root of $g$ to a given group $G$, where $n$ is a nonzero natural number and $g$ is an element of $G$ of infinite order, we compute the profinite completion $\widehat{G'}$ of $G'$. Also, given $G$ a profinite group in which any subgroup of finite index is open, $n$ a nonzero natural number, $g$ an element of $G$, and $x$ an element not belonging to $G$, we point out necessary and sufficient condition under which the group obtained by adjoining roots to the profinite group $G$ remains again profinite. Our proofs make use of theoretico-combinatorial methods.

A Note on Finite Nilpotent Groups

It is well known that if G is a group and H is a normal subgroup of G of finite index k, then x k ∈ H for every x ∈ G . We examine finite groups G with the property that x k ∈ H for every subgroup H of G, where k is the index of H in G. We prove that a finite group G satisfies this property if and only if G is nilpotent.

Profinite non-rigidity of arithmetic groups

We show that for a typical higher-rank arithmetic lattice \Gamma , there exist finite index subgroups \Gamma_{1} and \Gamma_{2} such that \Gamma_{1}\not\cong \Gamma_{2} while \widehat{\Gamma_{1}}\cong \widehat{\Gamma_{2}} . But there are exceptions to that rule.

Torsion Groups with the Norm of pd-Subgroup of Finite Index

The authors study the relations between the properties of torsion groups and their norms of $pd$-subgroups. The norm $N_G^{pdI}$ of $pd$-subgroups of a group $G$ is the intersection of the normalizers of all its $pd$-subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of $pd$-subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a finite extension of its norm of $pd$-subgroups if and only if it is a finite extension of its center. By this fact and the structure of the norm of $pd$-subgroups, we get that any torsion group that is a finite extension of this norm is locally finite.

Virtual endomorphisms of the group $pg$

A virtual endomorphism of a group $G$ is a homomorphism of the form $\phi:H\rightarrow G$, where $H<G$ is a subgroup of finite index. A virtual endomorphism $\phi:H\rightarrow G$ is called simple if there are no nontrivial normal $\phi$-invariant subgroups, that is, the $\phi$-core is trivial. We describe all virtual endomorphisms of the plane group $pg$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $pg$ admits a transitive self-similar (as well as finite-state) action of degree $d$ if and only if $d\geq 2$ is not an odd prime, and admits a self-replicating action of degree $d$ if and only if $d\geq 6$ is not a prime or a power of $2$.

Fusion and Positivity in Chiral Conformal Field Theory

In this article we show that the conformal nets corresponding to WZW models are rational, resolving a long-standing open problem. Specifically, we show that the Jones-Wassermann subfactors associated with these models have finite index. This result was first conjectured in the early 90s but had previously only been proven in special cases, beginning with Wassermann’s landmark results in type A. The proof relies on a new framework for the systematic comparison of tensor products (a.k.a. ‘fusion’) of conformal net representations with the corresponding tensor product of vertex operator algebra modules. This framework is based on the geometric technique of ‘bounded localized vertex operators,’ which realizes algebras of observables via insertion operators localized in partially thin Riemann surfaces. We obtain a general method for showing that Jones-Wassermann subfactors have finite index, and apply it to additional families of important examples beyond WZW models. We also consider applications to a class of positivity phenomena for VOAs, and use this to outline a program for identifying unitary tensor product theories of VOAs and conformal nets even for badly-behaved models.

Frattini subgroups of hyperbolic-like groups

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group G admitting a general type action on a hyperbolic space S, we show that the induced action of the Frattini subgroup Φ(G) on S has bounded orbits. This implies that Φ(G) is “small” compared to G; in particular, |G:Φ(G)|=∞. In contrast, for any finitely generated non-cyclic group Q with Φ(Q)={1}, we construct an infinite lacunary hyperbolic group L such that L/Φ(L)≅Q; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups.

Integrals of groups. II

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study:A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2).The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1).Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups (Section 5).The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class (Sections 6 and 7).Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1).Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2).We end the paper with a number of open problems.

Uncountable groups with small commutator subgroup

This paper continues the investigation of the behavior of the commutator subgroup of an uncountable group whose large subgroups have finite (bounded) index in their normal closures. Moreover, uncountable groups in which all large subgroups have small index in their normal closures are considered.

Maximal subgroups of a family of iterated monodromy groups

Abstract The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$ -generated weakly branch, but not branch, group that acts on the $m$ -adic tree, for $s,m\ge 2$ . A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

$\alpha$-induction for bi-unitary connections

The tensor functor called \alpha -induction produces a new unitary fusion category from a Frobenius algebra object, or a Q -system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor N\subset M of finite index and finite depth, which gives a braided fusion category of endomorphisms of N . It is also understood in terms of Ocneanu’s graphical calculus. We study this \alpha -induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4 -tensors appearing in recent studies of 2 -dimensional topological order. We show that the resulting \alpha -induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a local Q -system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented.