The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of imes0-products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams. result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
This paper introduces cluster exchange groupoids for Coxeter-Dynkin diagrams and finds their fundamental groups are braid groups.
problem Understanding the fundamental groups of cluster exchange groupoids for Coxeter-Dynkin diagrams.
method Introduced cluster exchange groupoids for Coxeter-Dynkin diagrams and showed the fundamental group isomorphic to braid groups.
result The fundamental group of the exchange groupoid for a Coxeter-Dynkin diagram is the braid group associated with the diagram.
This paper presents a construction of fibered links (K,Σ) out of chord diagrams $\sL$. Let Γ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of (K,Σ) equals the bilinear form of the simply-laced Coxeter system (W,S) associated to Γ; and the monodromy of $(K,Σ)…
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 …
Study of Coxeter diagrams and Artin-Tits groups, focusing on normalisers and wall intersections.
problem Understanding normalisers of parabolic subgroups in Artin-Tits groups and their connections to Coxeter diagrams.
method Analyzing hyperplane arrangements, Coxeter groups, and wall-and-chamber structures.
result Complexified hyperplane complement is a K(π,1) space for normalisers of parabolic subgroups in finite-type Coxeter diagrams.
Study on divergence and thickness for Coxeter groups, generalizing previous work.
problem Characterizing and bounding divergence and thickness for Coxeter groups.
method Characterization of linear divergence, introduction of hypergraph index, new construction of Coxeter systems.
result Upper bounds on divergence and thickness for Coxeter groups, conjectured to be equalities.
Study proves meridional rank conjecture for certain complex links.
problem Determining the minimum number of generators needed for link groups.
method Using Coxeter quotients and Wirtinger numbers, the study provides both lower and upper bounds.
result Proved meridional rank conjecture for specific types of links.
Reduces conjecture for Artin groups to simpler cases.
problem Proving K(π,1) for Artin groups with specific spherical parabolics. method Reduces to simpler cases, uses injective metric spaces, combinatorial convexity, and Bestvina-type inequalities.
result Deduces K(π,1) conjecture for specific Artin groups. A process enumerates rack elements from a presentation.
problem Systematically enumerating elements of racks from presentations.
method Generalizes Todd-Coxeter process for cosets, adapted for racks.
result Process terminates if and only if rack is finite, outputting operation tables.
Special covers of alternating links have finite index subgroups in certain groups.
problem Understanding the structure of alternating link complements and their subgroups.
method Constructing special covers with bounded degree and embedding into specific groups.
result Explicit bounds on the index of subgroups in right-angled Artin and Coxeter groups.
New construction shows DAAG/DAHA actions on congruence subgroups.
problem Understanding DAAG/DAHA actions on congruence subgroups.
method Coxeter-type presentations and adjoint DAAG/DAHA.
result DAAG/DAHA actions on congruence subgroups Γ1(r). We prove that all immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and the k-symmetric space structure on G/T i…
We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons exist only for n=3,4,6. We classify compact polygons which admit a transitive gr…
Reduces conjecture to tree-based Artin groups.
problem Proving K(π,1)-conjecture for all Artin groups. method Actions on Bestvina complexes of Garside groupoids.
result New classes of Artin groups satisfying the conjecture.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid gr…
HZ transform applied to knot polynomials reveals hyperbolic knot structures.
problem Understanding the structure of knot polynomials and their factorisability.
method Applying the Harer-Zagier transform to knot polynomials and character expansions.
result Construction of an infinite family of hyperbolic knots and proof of factorisability in the 3-strand case.
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 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…
Geometrically classifies total stability spaces for Dynkin diagrams.
problem Classifying total stability spaces for triangulated categories.
method Constructing a geometric model of root categories as hQ-gons and proving isomorphisms. result Total stability spaces ToStDb(Q)/[2] are isomorphic to moduli spaces of stable hQ-gons. 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.
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.
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.
Study Coxeter groups with maximal flats, finds dimension bound.
problem Understanding Coxeter groups with specific geometric properties.
method Analyzes relatively hyperbolic Coxeter groups of type HM with maximal subgroups of codimension 1.
result Bounded dimension of the groups studied.
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.
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.
Study confirms conjecture on Coxeter polyhedra growth rates in hyperbolic 3-space.
problem Growth rates of reflection groups in hyperbolic Coxeter polyhedra.
method Proved conjecture for ideal Coxeter polyhedra in H3; found smallest growth rate; explored correlations between volumes and growth rates. result Growth rates of ideal Coxeter polyhedra in H3 are always Perron numbers. 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.
Flat coordinates found for algebraic Frobenius manifolds in low dimensions.
problem Understanding algebraic Frobenius manifolds in small dimensions.
method Using reflection representations of finite Coxeter groups, finding flat coordinates of the Frobenius metric.
result Explicit relations between flat coordinates of the Frobenius metric and intersection form for most known examples up to dimension 4.
Study on right-angled Coxeter groups and their geometric properties.
problem Characterizing the coarse geometry of right-angled Coxeter groups.
method Analyzing graph properties and applying geometric group theory.
result Proves properties of right-angled Coxeter groups, including quasi-isometry and divergence.
The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-…
We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pse…
The paper calculates the Saito determinant for Coxeter discriminant strata.
problem Calculating the Saito determinant for specific geometric strata.
method Using the Saito flat metric and Lie derivatives, the paper finds the determinant of the metric restricted to Coxeter discriminant strata.
result The determinant of the Saito metric on Coxeter discriminant strata is proportional to a product of linear factors in flat coordinates.
New invariant classifies right-angled Coxeter groups, bounds thickness.
problem Classifying and bounding right-angled Coxeter groups.
method Introducing hypergraph index from defining graph, computing upper bounds.
result Hypergraph index partitions groups into quasi-isometry classes, bounds thickness.
New classification of hyperbolic Coxeter prisms.
problem Classifying hyperbolic Coxeter prisms.
method Determine which prisms are quasi-arithmetic or arithmetic.
result New insights into commensurability and systoles of associated orbifolds.
Surprising circles found in Coxeter group boundaries.
problem Embedded circles in Morse boundaries of Coxeter groups.
method Analysis of Morse boundaries and defining graphs.
result Circles not arising from visible Fuchsian subgroups.