Trivial Center Research Articles

Number-theoretic positive entropy shifts with small centralizer and large normalizer

AbstractHigher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

Character rigidity of simple algebraic groups

We prove the following extension of Tits’ simplicity theorem. Let $$k$$ be an infinite field, G an algebraic group defined and quasi-simple over $$k,$$ and $$G(k)$$ the group of $$k$$ -rational points of G. Let $$G(k)^+$$ be the subgroup of $$G(k)$$ generated by the unipotent radicals of parabolic subgroups of G defined over $$k$$ and denote by $$PG(k)^+$$ the quotient of $$G(k)^+$$ by its center. Then every normalized function of positive type on $$PG(k)^+$$ which is constant on conjugacy classes is a convex combination of $$\mathbf {1}_{PG(k)^+}$$ and $$\delta _e.$$ As corollary, we obtain that, when $$k$$ is countable, the only ergodic IRS’s (invariant random subgroups) of $$PG(k)^+$$ are $$\delta _{PG(k)^+}$$ and $$\delta _{\{e\}}.$$ A further consequence is that, when $$k$$ is a global field and G is $$k$$ -isotropic and has trivial center, every measure preserving ergodic action of $$G(k)$$ on a probability space either factorizes through the abelianization $$G(k)_{\mathrm{ab}}$$ or is essentially free.

Centers of subgroups of big mapping class groups and the Tits alternative

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a subgroup of some big mapping class group.

Connective Bieberbach groups

We prove that a Bieberbach group with trivial center is not connective and use this property to show that a Bieberbach group is connective if and only if it is poly-[Formula: see text].

On Thompson’s Conjecture for Finite Simple Exceptional Groups of Lie Type

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group.

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

The Normalizer Property for Integral Group Rings of Holomorphs of Finite Groups with Trivial Center

Let G = Hol(H) be the holomorph of a finite group H. If there is a prime q dividing ⋎H⋎ such that every q-central automorphism of H is inner and Z(H) = 1, then every Coleman automorphism of G is inner. In particular, the normalizer property holds for G.

Commutation semigroups of finite metacyclic groups with trivial centre

We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented $$\begin{aligned} G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in {\mathbb {Z}}^+) \end{aligned}$$with $$(m,k - 1) = 1$$ and $$n = \mathrm {ind}_m(k),$$ the smallest positive integer for which $${k^n} = 1\,\pmod m,$$ with the conjugate of a by b written $${a^b} = {b^{ - 1}}ab.$$ The right and left commutation semigroups ofG, denoted $$\mathrm{P}(G)$$ and $$\Lambda (G),$$ are the semigroups of mappings generated by $$\rho (g):G \rightarrow G$$ and $$\lambda (g):G \rightarrow G$$ defined by $$(x)\rho (g) = [x,g]$$ and $$(x)\lambda (g) = [g,x],$$ where the commutator of g and h is defined as $$[g,h] = {g^{ - 1}}{h^{ - 1}}gh.$$ This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to G, a metacyclic group with trivial centre. We give a construction of $$\mathrm{P}(G)$$ and $$\Lambda (G)$$ as unions of containers, an idea presented in the previous paper on dihedral groups. In the case that $$\left\langle a \right\rangle$$ is cyclic of order p or $${p^2}$$ or its index is prime, we show that both $$\mathrm{P}(G)$$ and $$\Lambda (G)$$ are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for G(m, n, k) with m prime, the condition $$\left| {\mathrm{P}(G)} \right| = \left| {\Lambda (G)} \right|$$ is equivalent to $$\mathrm{P}(G) = \Lambda (G).$$

On the cohomology class of fiber-bunched cocycles on semi simple Lie groups

We study the cohomological equation associated to linear cocycles on semi simple Lie groups \mathcal G over hyperbolic dynamics. We give sufficient conditions for the solution of the cohomological equation of fiber-bunched cocycles to be unique and for the Hölder conjugacy class of the cocycle to coincide with C^\nu(M,\mathcal G) . In particular, we prove that there exists an open and dense subset of the set C_b^\nu(M,\mathcal G) of fiber-bunched cocycles with trivial centralizer. As a consequence we deduce that the solutions of the {cohomological} equation of fiber-bunched cocycles form a finite abelian subgroup of C_b^\nu(M,\mathcal G) for an open and dense subset of fiber-bunched cocycles in C_b^\nu(M,\mathcal G) . Some results on the centralizer of skew-products are also given.

The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten theory

In this work, for a finite group G and a 4-cocycle ω∈Z4(G,k×), we compute explicitly the center of the monoidal 2-category 2VecGω of ω-twisted G-graded 1-categories of finite dimensional k-vector spaces. This center gives a precise mathematical description of the topological defects in the associated 3+1D Dijkgraaf-Witten TQFT. We prove that this center is a braided monoidal 2-category with a trivial sylleptic center.

Typical entanglement entropy in the presence of a center: Page curve and its variance

In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P(S_A) \mathrm{d}S_A$ that a subsystem has entanglement entropy $S_A$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, $\Delta S_A/\langle{S_{A}}\rangle\ll 1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.

On a Finite Group with Restriction on Set of Conjugacy Classes Size

The greatest power of a prime p dividing the natural number n will be denoted by $$n_p$$ . For a set of primes $$\pi $$ and a natural number n we will denote $$n_{\pi }=\prod _{p\in \pi }n_p$$ . Let G be a finite group with trivial center, and $$p,q>5$$ be distinct prime divisors of |G|. We prove that if for every nonunity conjugacy classes size $$\alpha $$ , it is true that $$\alpha _{\{p,q\}}\in \{p^n,q^m,p^nq^m\}$$ , where n and m depend only on p and q, then $$|G|_{\{p,q\}}=p^nq^m$$ , and $$C_G(g)\cap C_G(h)=1$$ for every p-element g and every q-element h.

On Thompson’s conjecture for finite simple groups

Let G be a finite group and N(G) be its set of conjugacy class sizes. We prove that if L is a finite simple non-abelian group and G is a finite group with trivial center such that then

Centralizers in the group of interval exchange transformations

We study the group of interval exchange transformations. Let T be an m-interval exchange transformation. By the rank of T we mean the dimension of the ℚ-vector space spanned by the lengths of the exchanged subintervals. We prove that if T satisfies Keane’s infinite distinct orbit condition and rank(T) > 1 + [m/2], then the only interval exchange transformations which commute with T are its powers. In the case that T is a minimal 3-interval exchange transformation, we prove a more precise result: T has a trivial centralizer in the group of interval exchange transformations if and only if T satisfies the infinite distinct orbit condition.

Anosov diffeomorphisms of products II. Aspherical manifolds

We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomorphisms. In the course of our study, we obtain a result of independent group theoretic and topological interest on the stability of the Hopf property, namely, that the product of finitely many Hopfian groups with trivial center is Hopfian.

Invariants of contact Lie algebras

The aim of this work is to prove that any contact Lie algebra has a semi-invariant. We explicitly compute it and give conditions for it to be an invariant of the contact Lie algebra. As a consequence we can prove that any perfect contact Lie algebra has trivial center. Moreover, we prove that a perfect Lie algebra is a contact Lie algebra if, and only if, it has only one invariant.

Artin groups of infinite type: Trivial centers and acylindrical hyperbolicity

While finite type Artin groups and right-angled Artin groups are well understood, little is known about more general Artin groups. In this paper we use the action of an infinite type Artin group A Γ A_{\Gamma } on a CAT(0) cube complex to prove that A Γ A_{\Gamma } has trivial center providing Γ \Gamma is not the star of a single vertex, and is acylindrically hyperbolic providing Γ \Gamma is not a join.

Bi-interpretability of some monoids with the arithmetic and applications

We will prove bi-interpretability of the arithmetic $${\mathbb {N}}= \langle N, +,\cdot , 0, 1\rangle $$ and the weak second order theory of $${\mathbb {N}}$$ with the free monoid $$\mathbb {M}_X$$ of finite rank greater than 1 and with a non-trivial partially commutative monoid with trivial center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable. Moreover, any recursively enumerable language in the alphabet X is definable in $$\mathbb {M}_X$$ . Primitive elements, and, therefore, free bases are definable in the free monoid. It has the so-called QFA property, namely there is a sentence $$\phi $$ such that every finitely generated monoid satisfying $$\phi $$ is isomorphic to $$\mathbb {M}_X$$ . The same is true for a partially commutative monoid without center. We also prove that there is no quantifier elimination in the theory of any structure that is bi-interpretable with $$\mathbb {N}$$ to any boolean combination of formulas from $$\Pi _n$$ or $$\Sigma _n$$ .

Classification of Osborn loops of order 4n

The smallest non-associative Osborn loop is of order 16. Attempts in the past to construct higher orders have been very difficult. In this paper, some examples of finite Osborn loops of order 4n, n = 4, 6, 8, 9, 12, 16 and 18 were presented. The orders of certain elements of the examples were considered. The nuclei of two of the examples were also obtained and these were used to establish the classification of these Osborn loops up to isomorphism. Moreover, the central properties of these examples were examined and were all found to be having a trivial center and no non-trivial normal subloop. Therefore, these examples of Osborn loops are simple Osborn loops.

Toeplitz subshifts with trivial centralizers and positive entropy

Given a dynamical system $(X,G)$, the centralizer $C(G)$ denotes the group of all homeomorphisms of $X$ which commute with the action of $G$. This group is sometimes called the automorphism group of the dynamical system $(X,G)$. In this note, we generalize the construction of Bulatek and Kwiatkowski (1992) to $\mathbb Z^d$-Toepltiz systems by identifying a class of $\mathbb Z^d$-Toeplitz systems that have trivial centralizers. We show that this class of $\mathbb Z^d$-Toeplitz with trivial centralizers contains systems with positive topological entropy.