Academic literature on the topic 'Twisted Artin group'

AbstractWe begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological calculations.

We consider Wada's representation as a twisted version of the standard action of the braid group,Bn, on the free group withngenerators. Constructing a free group,Gnm, of ranknm, we compose Cohen's mapBn→Bnmand the embeddingBnm→Aut(Gnm)via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation andm−1copies of the standard representation after changing the parameterttotkin the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group.

The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev : Bn →Bn, [Formula: see text], defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation τ : x ↦ Δ-1xΔ by the generalized half-twist (Garside element). More generally, the involution rev is defined for all Artin groups (equipped with Artin's presentation) and the involution τ is defined for all Artin groups of finite type. A palindrome is an element invariant under rev. We study palindromes and palindromes invariant under τ in Artin groups of finite type. Our main results are the injectivity of the map [Formula: see text] in all finite-type Artin groups, the existence of a left-order compatible with rev for Artin groups of type A, B, D, and the existence of a decomposition for general palindromes. The uniqueness of the latter decomposition requires that the Artin groups carry a left-order.

Die Fächerflügler (Strepsiptera) sind mit nur ca. 600 beschriebenen rezenten Arten eine kleine, parasitische Gruppe der holometabolen Insekten. Fossilfunde sind selten, aber in den letzten Jahren hat sich die Kenntnis der Stammgruppe der Strepsiptera durch die Entdeckung gut erhaltener Arten aus kreidezeitlichem burmesischem Bernstein und eozänem baltischen Bernstein stark vermehrt. Bis auf ganz wenige Ausnahmen, wie eine fossile Primärlarve aus burmesischem Bernstein und ein spätes weibliches Larvenstadium der †Mengeidae aus baltischem Bernstein, sind nur Männchen bekannt. Diese Bernsteinfossilien haben wesentlich zum Verständnis der Evolution der Strepsiptera im späten Mesozoikum und Känozoikum beigetragen. Die Stammgruppenvertreter der Fächerflügler werden vorgestellt und in einen evolutiven Kontext eingeordnet. The stem-group of the twisted-winged parasites (Insecta, Strepsiptera) Abstract: With only about 600 described extant species, the twisted-winged parasites (Strepsiptera) are a small, parasitic group of holometabolous insects. Fossil records of Strepsiptera are rare, but in the last years the knowledge of the stem group has greatly increased with the discovery of well-preserved species from Cretaceous Burmese amber and Eocene Baltic amber. With very few exceptions, such as a fossil primary larva from Burmese amber and a late female larval stage of the †Mengeidae from Baltic amber, only males are known. These amber fossils have greatly contributed to the understanding of the evolution of Strepsiptera in the late Mesozoic and Cenozoic. The stem group representatives of the twisted-winged parasites are described and placed in an evolutionary context.

AbstractLet$E_{/{\mathbb{Q}}}$be a semistable elliptic curve, andp≠ 2 a prime of bad multiplicative reduction. For each Lie extension$\mathbb{Q}$FT/$\mathbb{Q}$with Galois groupG∞≅$\mathbb{Z}$p⋊$\mathbb{Z}$p×, we constructp-adicL-functions interpolating Artin twists of the Hasse–WeilL-series of the curveE. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the$\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.

AbstractWe show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$. As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.

AbstractWe observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$-theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

Questa tesi consiste in tre capitoli principali, e tutti si evolvono intorno ai gruppi di Artin. Dimostrare risultati per tutti i gruppi di Artin è una sfida seria, quindi di solito ci si concentra su particolari sottoclassi. Tra le sottofamiglie più conosciute dei gruppi di Artin c'è la famiglia dei gruppi Artin ad angolo retto (RAAGs in breve). Si possono definire usando i grafici simpliciali, che determinano il gruppo fino all'isomorfismo. Sono anche interessanti perché ci sono una varietà di metodi per studiarli, provenienti da diversi punti di vista, come la geometria, l'algebra e la combinatoria. Questo ha portato alla comprensione di molti problemi dei RAAG, come il problema delle parole, la crescita sferica, le intersezioni di sottogruppi parabolici, ecc. Nel Capitolo 2 ci concentriamo sulla crescita geodetica dei RAAG, su grafi link-regolari, ed estendiamo un risultato in quella direzione, fornendo una formula della crescita su grafi link-regolari senza tetraedri. Nel capitolo 3 lavoriamo con gruppi leggermente diversi, la classe dei gruppi Artin contorti ad angolo retto (tRAAGs in breve). Sono definiti usando grafi misti, che sono grafi semplici in cui i bordi possono essere diretti. Troviamo una forma normale per presentare gli elementi in un tRAAG. Se dimentichiamo le direzioni dei bordi, otteniamo un grafo non diretto sottostante, che chiamiamo grafo ingenuo. Sul grafo ingenuo, che è semplice, si può definire un RAAG, che corrisponde naturalmente al nostro tRAAG. Discuteremo alcune somiglianze e differenze algebriche e geometriche tra i tRAAG e i RAAG. Usando la forma normale siamo in grado di concludere che la crescita sferica e geodetica di un tRAAG concorda con la rispettiva crescita del RAAG sottostante. Il capitolo 4 ha un tema diverso, e consiste nello studio dei sottogruppi parabolici nei gruppi pari di Artin. Il lavoro è motivato dai risultati corrispondenti nei RAAG, e generalizziamo alcuni di questi risultati a certe sottoclassi di gruppi pari di Artin. ​
This thesis consists of three main chapters, and they all evolve around Artin groups. Proving results for all Artin groups is a serious challenge, so one usually focuses on particular subclasses. Among the most well-understood subfamilies of Artin groups is the family of right-angled Artin groups (RAAGs shortly). One can define them using simplicial graphs, which determine the group up to isomorphism. They are also interesting as there are a variety of methods for studying them, coming from different viewpoints, such as geometry, algebra, and combinatorics. This has resulted in the understanding of many problems in RAAGs, like the word problem, the spherical growth, intersections of parabolic subgroups, etc. In Chapter 2 we focus on the geodesic growth of RAAGs, over link-regular graphs, and we extend a result in that direction, by providing a formula of the growth over link-regular graphs without tetrahedrons. In Chapter 3 we work with slightly different groups, the class of twisted right-angled Artin groups (tRAAGs shortly). They are defined using mixed graphs, which are simplicial graphs where edges are allowed to be directed edges. We find a normal form for presenting the elements in a tRAAG. If we forget about the directions of edges, we obtain an underlying undirected graph, which we call the naïve graph. Over the naïve graph, which is simplicial, one can define a RAAG, which corresponds naturally to our tRAAG. We will discuss some algebraic and geometric similarities and differences between tRAAGs and RAAGs. Using the normal form we are able to conclude that the spherical and geodesic growth of a tRAAG agrees with the respective growth of the underlying RAAG. Chapter 4 has a different theme, and it consists of the study of parabolic subgroups in even Artin groups. The work is motivated by the corresponding results in RAAGs, and we generalize some of these results to certain subclasses of even Artin groups. ​