The paper defines and explores Coxeter type LOTs for ribbon 2-knots.
problem Validity of Whitehead's asphericity conjecture for Ribbon 2-knots.
method Definition and analysis of Coxeter type LOTs and their groups.
result Existence of prime LOTs of Coxeter type with specified rank.
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 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.
A classical result of H. S. M. Coxeter asserts that a certain quotient B(m,n) of the braid group B(m) on m strands is finite if and only if (m,n) corresponds to the type of one of the five Platonic solids. If k is a knot or virtual knot, one can study similar quotients G(k,n) for the correspond…
Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös-Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about t…
The paper studies Coxeter quotients of surface braid groups.
problem Analyzing Coxeter quotients of surface braid groups.
method Analyzing Coxeter-type quotients of surface braid groups by normal closure and commutator relations.
result Characterized Coxeter quotients of surface braid groups.
Computes a specific homology for a type of braid.
problem Calculating a homology for a particular class of braids.
method Uses Khovanov-Rozansky homology for Coxeter braids on 4 strands.
result Computed the triply graded Khovanov-Rozansky homology.
We study relatively hyperbolic Coxeter groups of type HM with maximal Euclidean Coxeter subgroups of codimension 1. Our main result in this paper is that the dimension of these groups is bounded above.
In the present paper, we build a bridge between Conway-Coxeter friezes and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomials attached to rational links by using Conway-Coxeter friezes. As an application one can give a complete invariant on Conway-Coxeter friezes o…
Study Coxeter groups over fusion rings and their geometric realisations.
problem Understanding Coxeter groups and their embeddings.
method Investigate faithful realisations and Vinberg systems.
result Induce embeddings of hyperplane complements.
New methods classify hyperbolic polytopes with up to 40 facets.
problem Classifying compact hyperbolic Coxeter polytopes with specific facet counts.
method New combinatorial method via point set order types.
result Proves existence of a compact hyperbolic Coxeter 29-polytope with at least 40 facets.
Two quandles from Coxeter groups studied, showing similarities in automorphism groups.
problem Understanding quandles from Coxeter groups and their automorphism groups.
method Defined two families of quandles from Coxeter quandles and Pin groups, analyzed their automorphism groups.
result Similarity in inner automorphism groups of the two quandles.
Finite volume Coxeter polytopes are quasiperfect and related to finite covolume reflection groups.
problem Characterizing finite volume Coxeter polytopes and their relation to reflection groups.
method Analyzing Coxeter polytopes and their volumes within Vinberg domains.
result Finite covolume reflection groups are characterized by the Vinberg domain.
We construct a family of right-angled Coxeter groups which provide counter-examples to questions about the stable boundary of a group, one-endedness of quasi-geodesically stable subgroups, and the commensurability types of right-angled Coxeter groups.
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.
Flat coordinates for Frobenius manifolds defined on the orbit space of a Coxeter group W are specified through a certain system of generators of W-invariant polynomials. In this note, starting from basic invariants proposed by M.Mehta, we calculate flat coordinates for the exceptional groups of type E_7 and E_8, leadin…
CAT(0) properties extended to a class of Shephard groups.
problem Extending CAT(0) properties to Shephard groups.
method Introduced new criteria for CAT(1) complexes and extended results for Coxeter groups to Shephard groups.
result Shephard groups with enough finite parabolic subgroups are CAT(0).
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.
Smooth deformation space of Coxeter polytopes proven for orderable orbifolds.
problem Proving smoothness of deformation space for Coxeter polytopes.
method Analyzing natural map into realization space.
result Deformation space of Coxeter 3-polytopes is smooth.
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 …
We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type B. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type B and prove that such an extension supports a unique linear Markov trace function. We then introduce the Fram…
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. 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 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.
Triangulates permutahedra for Coxeter groups, revealing braid group connections.
problem Triangulating permutahedra for Coxeter groups.
method Constructs triangulations using total linear stability and height functions.
result Explicitly relates two braid group presentations.
We introduce a construction turning some Coxeter and Davis realizations of buildings into systolic complexes. Consequently groups acting geometrically on buildings of triangle types distinct from (2,4,4), (2,4,5), (2,5,5), and various rank 4 types are systolic.
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…
In this paper we classify all Markov traces on Iwahori-Hecke algebras associated with the finite Coxeter groups of type B. We then use these traces for constructing Jones-type invariants for oriented knots inside a solid torus, and finally we give skein-theoretical interpretations.
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.
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.
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.
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.
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.
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. 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.
We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cu…
Study complex reflections in infinite Coxeter tetrahedron moduli space.
problem Characterize representations of Coxeter group in complex hyperbolic space.
method Type-preserving representations of Coxeter group G to PU(3,1), parameterized by θ. result Discrete and faithful representations for θ∈[65π,π]. First nontrivial moduli space in complex hyperbolic space. 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.
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-…
The paper studies fibering properties of RACGs and random subcomplexes of buildings.
problem Higher virtual algebraic fibering properties of right-angled Coxeter groups.
method Generalization of Bestvina-Brady discrete Morse theory applied to Davis complex, combined with probabilistic arguments.
result Commutator subgroups of RACGs with certain finite building flag complexes admit epimorphisms to Z with strong topological finiteness properties.
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.
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 noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use…