Study real projective structures on a specific Coxeter orbifold.
problem Characterize real projective structures on a noncompact Coxeter orbifold.
method Embedding and extending a Coxeter quadrilateral, perturbing to form a convex polytope, and analyzing the deformation space.
result Determine the detailed properties of the deformation space of real projective structures on the orbifold.
Polytopes connect Lie theory to physics, integrating integrable systems.
problem Understanding connections between Lie theory and field theories.
method Using Coxeter Plane and integrable systems, a systematic mathematical treatment.
result Supports physical proposals linking polytopes to field theories.
A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to M is a noncompact complete hyperbolic surface Σ. We study double extensions of π1(M)≅π1(Σ) when Σ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…
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. Among all torus links, we characterise those arising as links of simple plane curve singularities by the property that their fibre surfaces admit only a finite number of cutting arcs that preserve fibredness. The same property allows a characterisation of Coxeter-Dynkin trees (i.e., An, Dn, E6, E7 and E8…
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 consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-…
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…
Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.
problem Finiteness of arithmetic maximal reflection groups in hyperbolic polyhedra.
method Observation of volume distribution and recent work with M. Fraczyk and S. Hurtado.
result Proof of finiteness of arithmetic maximal reflection 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.
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.
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.
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.
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…
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-…
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 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.
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 consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for c…
The paper explores growth rates and Perron numbers in Coxeter systems with low-dimensional Davis complexes.
problem Investigating growth rates and specific types of numbers in Coxeter systems with Davis complexes of low dimension.
method Examining Coxeter systems with Davis complexes of dimension at most 2, focusing on growth rates and specific types of numbers.
result The growth rate of Coxeter systems with Davis complexes of dimension at most 2 are either Salem or Pisot numbers, depending on the Euler characteristic.
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.
In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in H3. We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3…
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…
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.
Abstract Coxeter groups have growth rates that are Perron numbers.
problem Understanding growth rates of Coxeter groups.
method Defined a class of Coxeter groups, ∞--spanned, and analyzed their growth rates. result For ∞--spanned Coxeter groups, geodesic growth rate strictly dominates word growth rate and appears to be a Perron number. An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties and explain some new algorithms that allow their efficient enumeration. Using this we give the symmetry groups of all i-hedrites and the minimal representative for each. We also review the link of 4-hed…
The paper studies deformation spaces of Coxeter truncation polytopes.
problem Understanding the geometric properties and deformations of Coxeter truncation polytopes.
method Analyzing Coxeter truncation polytopes and their deformation spaces.
result Description of deformation spaces for Coxeter truncation polytopes of dimension d⩾4. New Coxeter groups have unique boundary structures.
problem Understanding boundaries of Coxeter groups.
method Recursive construction and amalgamation of CAT(0) groups.
result Totally disconnected Morse boundaries for new Coxeter groups.
Maximal rank Coxeter quotients found for 1.7M knots up to 16 crossings.
problem Finding maximal rank Coxeter quotients for knots.
method Computational approach to find Coxeter quotients for knots up to 16 crossings.
result Verification of Meridional Rank Conjecture for 595,515 knots.
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.
Outer automorphism groups of Coxeter groups are trivial except for small ranks.
problem Triviality of outer automorphism groups of Coxeter groups.
method Using Guirardel-Levitt outer space for free products, proving triviality and cyclic order for specific ranks.
result Outer automorphism groups are trivial except for small ranks.
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.
Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…
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.
Beside simplices, n-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n-cubes are not classified. We show that there is no hyperbolic Coxeter n-cube for n≥ 6, and provide a full classification for n≤5. Our methods, which are essentially of combin…
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.