The study of symmetries in manifolds derived from colored polytopes.
problem Existence and types of symmetries in rational homology 3-spheres.
method Analysis of hyperbolic manifolds and right-angled polytopes.
result Described how to create colorings with specific symmetries.
Locally rigid groups from 5-polytopes with Fuchsian ends.
problem Constructing locally rigid right-angled Coxeter groups with Fuchsian ends.
method Constructing a right-angled 5-polytope P and analyzing its right-angled Coxeter groups.
result Locally rigid right-angled Coxeter groups with Fuchsian ends can be constructed.
We prove that among four-dimensional ideal right-angled hyperbolic polytopes the 24-cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.
By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.
Study of infinitesimal rigidity in hyperbolic manifolds.
problem Proving infinitesimal rigidity of geometrically infinite hyperbolic manifolds.
method Developed a strategy to study infinitesimal rigidity of cyclic coverings of manifolds colored by right-angled polytopes.
result Proved infinitesimal rigidity of some hyperbolic 4- and 5-manifolds.
The paper sets new limits on hyperbolic polyhedra volumes.
problem Finding upper bounds on volumes of hyperbolic polyhedra.
method Analyzes three types of polyhedra: ideal, compact with finite vertices, and finite volume with mixed vertices.
result Establishes new upper bounds for polyhedra volumes in hyperbolic space.
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.
A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3…
New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.
problem Finding hyperbolic manifolds that fiber algebraically in all dimensions 5 to 8.
method Assigning colors and states to right-angled hyperbolic polytopes and applying arguments from Jankiewicz et al.
result First examples of hyperbolic manifolds with finitely presented but not of finite type fundamental groups.
By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.
We prove that every complete finite-volume hyperbolic 3-manifold M that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold W, which is also tessellated into right-angled regular pol…
The study constructs links from polytope subgraphs and proves their hyperbolic properties.
problem Proving hyperbolic structures for links from polytope subgraphs.
method Construction of 3-manifolds from polytope subgraphs and analysis of their topology.
result Hyperbolic links are parametrized by specific subgraphs in hyperbolic polytopes.
New Thurston norm defined for a specific type of groups using L2-invariants.
problem Measuring splitting complexity of integral characters in coherent right-angled Artin groups.
method Defining splitting complexity via L2-Euler characteristic and using Friedl--Lück's L2-polytope. result A Thurston-type semi-norm defined for measuring splitting complexity of integral characters.
The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.
problem Understanding the topology and geometry of simple orbifolds.
method Generalizing concepts from simple polytopes to simple orbifolds, focusing on simple handlebodies.
result Characterization of orbifold-aspherical properties and the existence of rank-two free abelian subgroups in terms of combinatorics.
In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H5/Γ where Γ is a torsion-free subgroup of minimal index of the congruence two subgroup Γ25 of the group Γ5 of positive units of the Lorentzian quadratic form x12+...+x52−x62. We also show that…
New method finds hyperelliptic 4-manifolds from polytope vector-colorings.
problem Finding hyperelliptic 4-manifolds from polytope vector-colorings.
method Introducing Hamiltonian subcomplexes and their corresponding subgroups.
result For dimensions ≤ 4, there is a bijection between Hamiltonian subcomplexes and hyperelliptic involutions.
Uniqueness of quasi-roots explored in right-angled Artin groups.
problem Uniqueness of quasi-roots in right-angled Artin groups.
method Introducing quasi-roots and studying their uniqueness.
result Uniqueness of quasi-roots established in right-angled Artin groups.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
problem Understanding the relationship between graph structures and algebraic properties of right-angled Artin groups.
method Analyzing the defining and extension graphs of right-angled Artin groups.
result Discovers connections to geometric group theory and complexity theory.
The paper examines rigidity in geometric actions of Coxeter groups on Croke-Kleiner spaces.
problem The rigidity of geometric actions of Coxeter groups compared to their quasi-isometric counterparts.
method Study of right-angled Coxeter groups acting geometrically on Croke-Kleiner spaces.
result Right-angled Coxeter groups have more rigid geometric actions than their quasi-isometric counterparts.
We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional…
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…
This paper characterizes a specific type of twisted Artin groups embedded in knot groups.
problem Embedding twisted right-angled Artin groups in knot groups.
method Defined and characterized twisted right-angled Artin groups through mixed graphs and Klein bottle relations.
result Completely determined which twisted right-angled Artin groups can be embedded in knot groups.
Right-angled Artin groups are classified based on measure equivalence.
problem Classifying right-angled Artin groups using measure equivalence.
method Proved measure equivalence implies isomorphic extension graphs, and used quasi-isometry results.
result No right-angled Artin group is superrigid for measure equivalence.
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…
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…
Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.
problem Finding bounds on ideal vertices in hyperbolic polyhedra.
method Improved Nikulin's inequality and Nonaka's lower bound.
result Shorter proofs and improved bounds on ideal vertices.
The study examines subgroups of RACGs and RAAGs, focusing on their RAAG properties.
problem Characterizing subgroups of right-angled Coxeter and Artin groups that are themselves RAAGs.
method Analyzes specific classes of subgroups and uses quasi-isometry and commensurability properties.
result Characterizes finite-index visual RAAG subgroups of 2-dimensional RACGs and provides new examples of RACGs commensurable to RAAGs.
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.
Proves Gromov's conjecture for a specific type of groups.
problem Gromov's conjecture for right-angled Artin groups.
method Analyzes universal covering spaces of manifolds with specific fundamental groups.
result Confirms Gromov's conjecture for right-angled Artin groups.
The study broadens the concept of cyclic polytopes to Veronese polytopes.
problem Extending the framework of cyclic polytopes to a broader class of polytopes.
method Described facial structure and combinatorial characterisation of facets via σ-parity alternating sequences.
result Established a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets.
Explicitly generates right-angled Artin subgroups from mapping classes.
problem Generating right-angled Artin subgroups from mapping classes.
method Explicit constant N depending on the collection of pure mapping classes, showing Nth powers generate the subgroup.
result Explicitly generated subgroups are undistorted.
We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed f…
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…
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …
Study on finiteness property of right-angled Artin groups actions on extension graphs.
problem Finiteness property of hyperbolic simplicial actions on right-angled Artin groups.
method Analysis of right-angled Artin group actions on extension graphs, using asymptotic translation lengths and syllable lengths.
result Asymptotic translation lengths of elements in right-angled Artin groups are rational and have a common denominator under certain conditions.
Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space Hn has at least one cusp for n≥5. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least th…
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 right-angled Artin subgroups found in Artin groups.
problem Finding large right-angled Artin subgroups in Artin groups.
method Examining centers of irreducible spherical special subgroups and their powers.
result Conjecture verified for certain classes of Artin groups, leading to hyperbolic surface subgroup conclusions.
Proves involutions on Right-angled Coxeter groups without fixed points.
problem Fixed-point-free involutions on group boundaries.
method Analyzes Right-angled Coxeter groups, proving conjecture variation.
result Proves involutions without fixed points on boundaries of Right-angled Coxeter groups.
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. Neural networks approximate unit spheres as polytopes.
problem Approximating unit spheres with neural networks.
method Using ReLU activation in neural networks to generate polytopes.
result Neural networks can approximate unit spheres as polytopes.
We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting…
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…
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.
Classifies divergence and thickness in right-angled Coxeter groups.
problem Characterizing the divergence and thickness of right-angled Coxeter groups.
method Completely classifies divergence functions and proves conditions for thickness using the hypergraph index.
result Exact divergence functions of RACGs can be computed from their defining graphs.