The study examines extensions of a projective plane minus two discs in flat affine Lorentzian 3-manifolds.
problem Classifying proper actions of affine Coxeter extensions on Minkowski space.
method Investigates double extensions of fundamental groups and their actions on Minkowski space.
result Identifies proper actions of the double extension that do not admit crooked fundamental domains.
The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…
Study on connectivity of Morse boundaries of Coxeter groups.
problem Connectivity of Morse boundaries of Coxeter groups.
method Defined conditions on defining graphs (wide-avoidant, wide-spherical-avoidant) and characterized Morse boundaries based on these conditions.
result Characterization of Morse boundary connectivity for different classes of Coxeter groups.
Constructs proper affine actions for right-angled Coxeter groups.
problem Proper affine actions for right-angled Coxeter groups.
method Constructs proper actions of right-angled Coxeter groups on O(p,q+1) and its Lie algebra by affine transformations.
result Any virtually special group admits proper affine actions on some R^n.
For a Coxeter group W we have an associating bi-linear form B on a real vector space. We assume that B has the signature (n−1,1). In this case we have the Cannon-Thurston map for W, that is, a W-equivariant continuous surjection from the Gromov boundary of W to the limit set of W. We focus on the case w…
We prove a long-standing conjecture about complex reflection arrangements.
problem The K(π,1) conjecture for affine Artin groups. method Recent advancements in dual Coxeter and Artin groups theory, new constructions, and poset shellability.
result The complexified complement of an affine reflection arrangement is a classifying space.
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …
Proof of K(π,1) conjecture for affine Artin groups.
problem Asphericality of complements of affine hyperplane arrangements.
method Combinatorics of noncrossing partition posets, dual Artin groups, and topological models.
result Affine Artin groups are aspherical.
The paper proves there are many Lagrangian fillings for Legendrian links of affine type.
problem Proving the existence of many Lagrangian fillings for Legendrian links of affine type.
method Using cluster structures and Coxeter mutation to prove the existence of fillings.
result There are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type.
Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.
problem Understanding knots created by Coxeter galleries.
method Examined knots in affine Coxeter complex of type \widewedge{B3}, constructing galleries and proving properties.
result Found bounds on stick number and smallest length of symmetric trefoils.
Explicit adjoint group description for Coxeter quandles.
problem Understanding the adjoint group structure of Coxeter quandles.
method Explicit descriptions and constructions of adjoint groups, using central extensions and 2-cocycles.
result The adjoint group of a Coxeter quandle is an intermediate group between the Coxeter group and its Artin group, with specific properties related to commutator subgroups and root systems.
The G-function associated to the semi-simple Frobenius manifold C^n/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-…
There exist natural generalizations of the real moduli space of Riemann spheres based on manipulations of Coxeter complexes. These novel spaces inherit a tiling by the graph-associahedra convex polytopes. We obtain explicit configuration space models for the classical infinite families of finite and affine Weyl groups …
Extended dual Coxeter and Artin groups theory to rank-three systems.
problem Extend dual Coxeter and Artin groups theory to rank-three systems.
method Geometric, combinatorial, and topological techniques.
result Proved the K(π,1) conjecture, triviality of the center, and solubility of the word problem for rank-three Artin groups. The paper proves conditions for the isomorphism between standard and dual Artin groups.
problem Conditions for the isomorphism between standard and dual Artin groups.
method Analyzes Coxeter systems and their actions on reduced words to prove isomorphisms.
result Proves conditions for the isomorphism between standard and dual Artin groups.
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
Groups with specific properties have similar cubulations and coarse median structures.
problem Understanding the structure of certain groups through cubical coarsening.
method Analyzing right-angled Artin and Coxeter groups, focusing on automorphisms and cubulations.
result Automorphisms of specific groups preserve coarse median structures and have nice fixed subgroups.
The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type A2, which descends to a fait…
Extends Framization to Coxeter system of type B.
problem Extending Framization to Coxeter systems of type B.
method Define a natural extension of the classical Temperley-Lieb algebra, prove the existence of a unique linear Markov trace function, introduce Framization as a quotient of the Yokonuma-Hecke algebra, and provide conditions for the Markov trace to pass to the quotient.
result Construct invariants for framed and classical links inside the solid torus.
Affine Artin groups have a finite classifying space.
problem Proving the K(π,1) conjecture for affine Artin groups. method Dual Garside structures, Euclidean isometries, and shellability of noncrossing partitions.
result Affine Artin groups have a finite classifying space.
The study finds many Lagrangian fillings for Legendrian links of specific types.
problem Understanding the number and types of Lagrangian fillings for Legendrian links.
method Proved the existence of at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite or affine Dynkin type.
result Found many Lagrangian fillings with rotational and conjugation symmetries for specific types of Legendrian links.
The study proves theorems about quasiflats in hierarchically hyperbolic spaces.
problem Understanding the structure of hierarchically hyperbolic groups.
method Proving theorems about quasiflats and hierarchically hyperbolic groups.
result Proves that a group is hyperbolic if it contains no Z2 subgroups and contains a uniform-quality quasiflat. New groups with Menger curve boundaries found.
problem Finding non-hyperbolic groups with specific boundary shapes.
method Using Coxeter groups with complete graph nerves and embedding theorems.
result First non-hyperbolic CAT(0) groups with Menger curve boundaries discovered. The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.
problem The congruence subgroup property for small Coxeter groups.
method Analyzes the properties of small Coxeter groups, proving the failure of the congruence subgroup property for certain groups.
result Proves the failure of the congruence subgroup property for infinite small Coxeter groups which are not virtually abelian.
An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.
Let W⋉L be an irreducible affine Weyl group with Coxeter complex Σ, where W denotes the associated finite Weyl group and L the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of Σ by the lattice L. We show that the ordinary and flag h-polynomial…
New combinatorial framework for geometric realizations of subword complexes.
problem Proving or disproving geometric realizations of subword complexes of Coxeter groups.
method Algebraic combinatorics and discrete geometry framework, parameter matrices.
result Existence of parameter matrices equivalent to realizability of subword complexes as chirotopes.
Study on random Coxeter groups, extending Erdös-Rényi model.
problem Understanding random Coxeter groups and their properties.
method Extended Erdös-Rényi model to study random general Coxeter groups.
result Results on homology of nerve and hyperbolicity of random Coxeter groups.
In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for n≥9 one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is tha…
New proof shows not all Salem numbers are growth rates of Coxeter groups.
problem Identifying growth rates of Coxeter groups using Salem numbers.
method New proof using spectral radii and Coxeter transformations.
result Not every Salem number is a growth rate of hyperbolic Coxeter groups.
Faces of quasi-arithmetic Coxeter polytopes are also quasi-arithmetic.
problem Characterizing faces of quasi-arithmetic Coxeter polytopes.
method Proof of quasi-arithmetic property of faces and sufficient condition for arithmetic faces.
result Lower-dimensional faces of quasi-arithmetic Coxeter polytopes are quasi-arithmetic.
In the present paper we define dual monoids for all Artin-Tits groups and we prove that for the type A~n we get a (quasi)-Garside structure. Such a structure provides normal forms for the Artin-Tits group elements and allows to solve some questions such as to determine the centralizer of a power of the Coxeter…
The study examines geometric conditions on CAT(0) cube complexes and Coxeter groups.
problem Classifying and understanding divergence in CAT(0) cube complexes and Coxeter groups.
method Geometric conditions on hyperplanes of CAT(0) cube complexes.
result Quadratic divergence for all right-angled Coxeter groups and no divergence between quadratic and cubic.
Survey on Coxeter groups for Lie group examples.
problem Understanding Coxeter groups and their applications.
method Constructing discrete subgroups of Lie groups using Coxeter groups.
result Coxeter groups provide new examples in discrete subgroups of Lie groups.
Paper connects Conway-Coxeter friezes and rational tangles.
problem Understanding the relationship between Conway-Coxeter friezes and rational tangles.
method Using Kauffman bracket polynomials to compute and connect friezes and rational tangles.
result Provides a complete invariant for Conway-Coxeter friezes of zigzag-type.
New proofs for growth series of Coxeter groups using complex structures.
problem Proving new formulae for growth series of Coxeter groups.
method Using the structure of Coxeter complexes, Davis complexes, or Tits non-complexes.
result Several classical formulae for growth series are proved in a new way.
Constructs Bach flat manifolds using modified Riemannian extension.
problem Constructing Bach flat manifolds of signature (2,2).
method Modified Riemannian extension of affine surfaces.
result Constructs scalar invariants not of Weyl type.
3D hyperbolic Coxeter groups' growth rates are Perron numbers.
problem Growth rates of 3D hyperbolic Coxeter groups.
method Combining Parry's result and \cite{Y}'s main result.
result Growth rates of 3D hyperbolic Coxeter groups are Perron numbers.
Study on Coxeter groups' boundary planarity, finding exceptions.
problem Planarity of Coxeter groups' boundaries under right angles.
method Characterization of defining graphs and analysis of boundaries.
result Non-planarity of defining graphs does not always imply non-planarity of boundaries.
New Coxeter groups yield n-dimensional Sierpiński boundaries.
problem Creating Coxeter groups with specific boundary shapes.
method Defined a class of right-angled Coxeter systems and provided conditions for their boundaries.
result Coxeter groups produce boundaries homeomorphic to n-dimensional Sierpiński compacta.
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
Characterizes Coxeter groups with convex cocompact representations in projective space.
problem Understanding representations of Coxeter groups as convex cocompact reflection groups.
method Investigates representations of Coxeter groups into GL(n,R) as geometric reflection groups in projective space.
result Characterizes Coxeter groups that admit convex cocompact representations and describes the spaces of such representations.
We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.
We classify hyperbolic Coxeter n-cubes for n≤5 and show none for n≥6.
problem Classifying hyperbolic Coxeter n-cubes for n≤5 and proving none for n≥6. method Combinatorial and algebraic methods implemented in Mathematica.
result No hyperbolic Coxeter n-cubes exist for n≥6. New groups found in hyperbolic 4D and 5D space have minimal growth rate.
problem Finding minimal growth rates in hyperbolic Coxeter groups.
method Combinatorial properties of hyperbolic Coxeter polyhedra, partial classification results, monotonicity properties of growth rates.
result Coxeter groups G4 and G5 in H4 and H5 have the smallest growth rate. The paper characterizes Conway-Coxeter friezes using rational links.
problem Characterizing Conway-Coxeter friezes of zigzag type.
method Characterization via rational links and application to Jones polynomial.
result Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type.
We prove that two angle-compatible Coxeter generating sets of a given finitely generated Coxeter group are conjugate provided one of them does not admit any elementary twist. This confirms a basic case of a general conjecture which describes a potential solution to the isomorphism problem for Coxeter groups.
Knot homology of Coxeter links identified with line bundles on Hilbert schemes.
problem Identifying knot homology for Coxeter links.
method Using line bundles on a generalized flag Hilbert scheme of points in \(\mathbb{C}^2\).
result Knot homology of Coxeter links corresponds to sections of a specific line bundle.