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.
Geometric constraints help classify hyperbolic polytopes.
problem Classifying reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
method Established geometric constraints on compact Coxeter polytopes in hyperbolic spaces.
result Geometric constraints are useful for classifying hyperbolic polytopes.
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.
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.
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.
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
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.
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.
Let Γ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph Γ is CFS, we prove that the right-angled Coxeter group GΓ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometr…
Study on connectivity and geometry of random Coxeter groups.
problem Connectivity threshold for square percolation on random graphs.
method Probabilistic combinatorics and techniques from geometric group theory.
result Determines connectivity threshold and cubical coarse median structure for random Coxeter groups.
New families of hyperbolic polyhedra yield infinitely many unique reflection groups.
problem Understanding commensurability classes of compact Coxeter polyhedra in hyperbolic spaces.
method Analyzing families of compact Coxeter polyhedra constructed by Makarov.
result Proves infinitely many commensurability classes in 4- and 5-dimensional hyperbolic spaces.
Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.
problem Characterize the geometric transitions of a Coxeter group's holonomy representations.
method Analysis of rigidity properties and character varieties in hyperbolic and Anti-de Sitter spaces.
result Description of singularity at the collapse of a right-angled cuboctahedron.
Minimal covolume group found in hyperbolic 3-space.
problem Finding groups with minimal covolume in hyperbolic 3-space.
method Proved existence of a specific group with minimal covolume.
result Minimal covolume group found with covolume equal to Catalan's constant.
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. In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with π/2 angles between reflect…
The boundary of certain hyperbolic groups is like a Menger curve.
problem Characterizing boundaries of hyperbolic Coxeter groups.
method Analyzing the nerve of hyperbolic right-angled Coxeter groups.
result Many triangulations and disks have boundaries homeomorphic to the Menger curve.
Associated to any Coxeter system (W,S), there is a labeled simplicial complex L and a contractible CW-complex ΣL (the Davis complex) on which W acts properly and cocompactly. ΣL admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of Sn−1,…
We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-c…
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. The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.
Explicitly constructed 5-manifolds tessellated by prisms.
problem Constructing closed arithmetic hyperbolic 5-manifolds.
method Explicit construction and tessellation of manifolds by Coxeter simplicial prisms.
result Explicit construction of 5-manifolds with specified properties.
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group …
Characterizes bi-Perron numbers with specific Galois conjugates.
problem Understanding bi-Perron numbers with real or unimodular conjugates.
method Characterization through power properties and spectral radii of transformations.
result Bi-Perron numbers with real or unimodular conjugates admit a power as stretch factors or spectral radii.
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.
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…
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.
A theorem of Tits - Vinberg allows to build an action of a Coxeter group Γ on a properly convex open set Ω of the real projective space, thanks to the data P of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically fi…
The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space W admits a faithful representation as a discrete reflection grou…
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.
We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer …
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 examines the systole of 3-manifolds with positive scalar curvature.
problem Analyzing the systole of 3-manifolds with positive scalar curvature.
method Local-to-global approach using capillary prisms and Coxeter gluing.
result Estimates the systole of 3-manifolds with positive scalar curvature.
We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus g > 1 is 24(g-1),…
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.
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…
This is a survey paper on the topic of Weil-Petersson geometry of Teichmuller spaces. Even though historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert ove…
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 moduli space of convex projective structures on a simplicial hyperbolic Coxeter orbifold is either a point or the real line. Answering a question of M. Crampon, we prove that in the latter case, when one goes to infinity in the moduli space, the entropy of the Hilbert metric tends to 0.
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.
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.