Bulletin of the London Mathematical Society最新文献

We prove that the lamplighter group admits an injective Lipschitz map to any finitely generated metabelian group that is not virtually nilpotent. This implies that finitely generated metabelian groups satisfy the “analytically thin/analytically thick” dichotomy recently introduced by Hume, Mackay, and Tessera.

The theory of Leavitt path algebras is intrinsically related, via graphs, to the theory of symbolic dynamics and $�C^{\ast }$-algebras where the major classification programs have been a domain of intense research in the last 50 years. In this article, we gather together current lines of research in the classification of Leavitt path algebras, questions, conjectures, and some of the results about them that have been obtained so far.

We show that a Fourier–Mukai equivalence between smooth projective varieties of characteristic $�p$ that commutes with either pushforward or pullback along Frobenius is a composition of shifts, isomorphisms, and tensor products with invertible sheaves whose $��(�p�-�1�)�$th tensor power is trivial.

We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

Let $�R$ be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if $�R$ is left quasi-Auslander, then $�R$ is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if $�R$ satisfies the Auslander condition, then $�R$ is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that $�R$ is Gorenstein if $�R$ satisfies the Auslander condition.

We use the degree of the colored Jones knot polynomials to show that the crossing number of a $��(�p�,�q�)�$-cable of an adequate knot with crossing number $�c$ is larger than $��q^2��c�$. As an application, we determine the crossing number of 2-cables of adequate knots. We also determine the crossing number of the connected sum of any adequate knot with a 2-cable of an adequate knot.

We show that the class of large-type Artin groups is invariant under isomorphism, in stark contrast with the corresponding situation for Coxeter groups. We obtain this result by providing a purely algebraic characterisation of large-type Artin groups (i.e. independent of the presentation graph). As a corollary, we completely describe the Artin groups isomorphic to a given large-type Artin group, and characterise those large-type Artin groups that are rigid.

Let $��E�/�Q�$ be an elliptic curve given by an integral Weierstrass equation. Let $��P�\in �E�\left(�Q�\right)�$ be a point of infinite order, and let $�{�\left(�B_n�\right)�}_{�n�⩾�1�}$ be the elliptic divisibility sequence generated by $�P$. This paper is concerned with a question posed in 2007 by Everest, Reynolds and Stevens: does $�{�\left(�B_n�\right)�}_{�n�⩾�1�}$ contain only finitely many perfect powers? We answer this question positively under the following three additional assumptions:

Let $�X$ be a random variable and define its concentration function by