arXiv research

Study real projective structures on a specific Coxeter orbifold.

Polytopes connect Lie theory to physics, integrating integrable systems.

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 $π_1 (M) \cong π_1 (Σ)$ 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.

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., $A_n$, $D_n$, $E_6$, $E_7$ and $E_8$…

Geometric models for Lie algebras from simple singularities.

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.

New proof shows not all Salem numbers are growth rates of Coxeter groups.

Survey on Coxeter groups for Lie group examples.

Study on connectivity of Morse boundaries of Coxeter groups.

New proofs for growth series of Coxeter groups using complex structures.

Introduces Coxeter polyhedra in various geometries.

Characterizes Coxeter groups with convex cocompact representations in projective space.

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.

New groups found in hyperbolic 4D and 5D space have minimal 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.

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…

New classification of hyperbolic Coxeter prisms.

Surprising circles found in Coxeter group boundaries.

The paper calculates the Saito determinant for Coxeter discriminant strata.

New noncompact Coxeter polytopes found in various dimensions.

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.

Study on divergence and thickness for Coxeter groups, generalizing previous work.

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 $\mathbb{H}^3$. 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.

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.

New Coxeter groups have unique boundary structures.

Maximal rank Coxeter quotients found for 1.7M knots up to 16 crossings.

Study Coxeter groups over fusion rings and their geometric realisations.

Computes a specific homology for a type of braid.

Characterizes Coxeter groups with specific boundary shapes.

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.

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…

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 $n\geq~6$, and provide a full classification for $n\leq 5$. Our methods, which are essentially of combin…

The study classifies all compact 5D polytopes with 9 facets.

The study classifies all compact hyperbolic polytopes with eight facets.

The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.

For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the …

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…