Artin groups and the fundamental groups of some moduli spaces

Configuration Lie groupoids and orbifold braid groups

On cohomology of crystallographic groups with cyclic holonomy of split type

On loxodromic actions of Artin–Tits groups

Throughout this work we explore new two-dimensional lattices constructed from the Archimedean tiling of the plane, characterizing its band structure and topological quantum spin Hall phases. As a proof of principle, within a combination of density functional theory and effective Hamiltonians, we have explored the material realization of Archimedean lattices. Fist, focusing in the kagome lattice, we predict within magnetic metal-organic frameworks the arising of quantum anomalous Hall effects. Additionally, from the time reversal symmetric counterpart, i.e. quantum spin Hall effect, we include a new multilayer degree of freedom into the control of metal-organic frameworks electronic properties. Within this multilayer system a layer localization effect of the topological states can be controlled through a external electric field. Focusing in a different Archimedean lattice, we shown its material realization in a two-dimensional boron allotrope. This material presents linear dispersive Dirac fermions in a px/py orbital pseudospin subspace. Here we shown a helical behavior of these orbital pseudospinors, i.e., its direction locked with the momentum direction, in a analogous way as the three dimensional topological insulator surface states. Furthermore, by adding a layer degree of freedom in addition to the orbital one, i.e., stacking of borophene bilayers, allows a control of the orbital texture by engineering the stacking order and external electric field. Lastly, we explored the construction of high-degeneracy points, which are not predicted by space-group symmetries. Here we show that pair permutation symmetries stabilizes such points, which are presents in a two Archimedean lattices, but also in other 3D lattices. We have found the recipe for constructing such lattices, where its experimental realization was proposed in metal lattices designed over silicon carbide surface oxidized by a ordered silica phase. In such systems, the post-transition metals s orbitals give rise to surface states localized neatly within the substrate’s semiconducting energy gap.

Introduction Around 1900 Hilbert, Hecke ([7]), Blumenthal ([2]) and others started the study of certain 2-dimensional complex spaces, which are closely related to the classification of special types of 2-dimensional abelian varieties. These complex spaces can easily be described. In fact, let n be a natural number, n > 1, which is square free and let K = Q q/n). Then, if o K is the ring of algebraic integers of K, the group SLz(oK) operates in a natural way on ~ • ~ and on ~ • ~ where ~ is the upper and ~ the lower half plane of C. The quotients of See • ~ and Jg x J g by the action of SL2(ov,) are the 2-dimensional complex spaces mentioned above. If the field K has a unit of negative norm, then the two actions on ~ • ~ and ~ • J g are isomorphic. This is true if n is a prime congruent 1 rood 4, the only case we shall consider in this paper. Therefore, from now on we assume that K= Q (l,/p), where p is a prime congruent 1 mod 4. The complex space Jg x .)~/SLz(oK) can be compactified by means of a finite number of points, called the cusps, to a compact 2-dimensional complex space. After resolving the cusps and also the quotient singularities on ~ x ~ / S L 2 (OK) , both in a canonical, explicit way, a nonsingular compact complex surface Y(R) is obtained, which in fact is an algebraic surface. The field of meromorphic (i. e. rational) functions on Y(p) is isomorphic to the field of meromorphic functions of • ~ / S L 2 (OK). On the other hand, although no complete classification of algebraic surfaces is known, there exists a rough classification in several classes (for most of which the surfaces contained in that class can be classified completely, at least in principle). In big outline this classification was already known to the italian school, but its precise formulation (this time also covering the non-algebraic case) and many of the proofs involved are due to Kodaira. Now the question considered in this paper is the following: where are the surfaces Y(p) to be placed in the rough classification of algebraic surfaces?

To a non-singular algebraic surface in space of five dimensions there can generally (the Veronese surface, of order 4, and cones are exceptional) be drawn, from an arbitrary point, a finite number of chords. If such a surface be projected from a point into space of four dimensions, there will, therefore, in general, be a certain number of points upon the resulting surface, at which two sheets of this surface, with distinct tangent planes, have an isolated common point. Such points have been called improper double points. We consider an algebraic surface ψ, in space of four dimensions [4], with no other multiple points than such double points, which we shall call accidental double points. The chords of the surface ψ, drawn from an arbitrary point O of the space [4], form a surface, or conical sheet, of which a general generator meets the surface in two points. The locus of these points is a curve which we shall call the chord curve. This curve has an actual double point at each of the accidental double points of ψ There will also, generally, be a certain number of points of the surface which are points of contact of tangent planes of the surface passing through O (and therefore also points of contact of tangent lines through O, these tangent lines being generally tangent lines of the chord curve).

Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form. More specifically, in the case when X is compact (namely projective), we need to exclude only the case when X is birationally equivalent to a K3 surface (a simply connected compact complex surface which admits a globally non-vanishing holomorphic 2-form) that is neither elliptic nor Kummer. With the exceptions noted above, we show that for any algebraic surface of Kodaira dimension less than 2, dominability by C^2 is equivalent to the apparently weaker requirement of the existence of a holomorphic image of C which is Zariski dense in the surface. With the same exceptions, we will also show the very interesting and revealing fact that dominability by C^2 is preserved even if a sufficiently small neighborhood of any finite set of points is removed from the surface. In fact, we will provide a complete classification in the more general category of (not necessarily algebraic) compact complex surfaces before tackling the problem in the case of non-compact algebraic surfaces.

Although in observational studies, propensity score matching is the most widely used balancing method, it has received much criticism. The main drawback of this method is that the individuals of the case and control groups are paired in the compressed one-dimensional space of propensity scores. In this paper, such a novel multivariate weighted k-nearest neighbours-based control group selection method is proposed which can eliminate this disadvantage of propensity score matching. The proposed method pairs the elements of the case and control groups in the original vector space of the covariates and the dissimilarities of the individuals are calculated as the weighted distances of the subjects. The weight factors are calculated from a logistic regression model fitted on the status of treatment assignment. The efficiency of the proposed method was evaluated by Monte Carlo simulations on different datasets. Experimental results show that the proposed Weighted Nearest Neighbours Control Group Selection with Error Minimization method is able to select a more balanced control group than the most widely applied greedy form of the propensity score matching method, especially for individuals characterized with few descriptive features.

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

Detecting homotopy equivalences in base-point-free homotopy

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2. We extend a well‐known result that Coxeter groups are to a class of Shephard groups that have ‘enough’ finite parabolic subgroups. We also show that in this setting, if the associated Coxeter group is type (FC), then the Shephard group acts properly and cocompactly on a cube complex. As part of our proof of the former result, we introduce a new criteria for a complex made of simplices to be .

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for which we get a complete orbifold or at least a Zariski open subset thereof, and we analyze these cases in some detail (e.g., we determine their orbifold fundamental group). In this set-up, the principal results of Deligne-Mostow on the Lauricella hypergeometric differential equation and work of Barthel-Hirzebruch-Höfer on arrangements in a projective plane appear as special cases. Along the way we produce in a geometric manner all the pairs of complex reflection groups with isomorphic discriminants, thus providing a uniform approach to work of Orlik-Solomon.

Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections.The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-difference operator in the algebra.For each irreducible representation of G the differential-difference equations lead to a linear system of first-order meromorphic differential equations corresponding to an integrable connection over the G-orbits of regular points in the complexification of the Euclidean space.The fundamental group is the generalized Artin braid group belonging to G, and its monodromy representation factors over the Hecke algebra of G. Monodromy has long been of importance in the study of special functions of several variables, for example, the hyperlogarithms of Lappo-Danilevsky are used to express the flat sections and the work of Riemann on the monodromy of the hypergeometric equation is applied to the case of dihedral groups.

Teichmüller Spaces and Mapping Class Groups