arXiv research

The study of symmetries in manifolds derived from colored polytopes.

Locally rigid groups from 5-polytopes with Fuchsian ends.

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.

The paper sets new limits on hyperbolic polyhedra volumes.

Study character varieties of a Coxeter group in hyperbolic and Anti-de Sitter spaces.

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.

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.

New Thurston norm defined for a specific type of groups using $L^2$-invariants.

The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form $H^5/Γ$ where $Γ$ is a torsion-free subgroup of minimal index of the congruence two subgroup $Γ^5_2$ of the group $Γ^5$ of positive units of the Lorentzian quadratic form $x_1^2+...+x_5^2-x_6^2$. We also show that…

New method finds hyperelliptic 4-manifolds from polytope vector-colorings.

Uniqueness of quasi-roots explored in right-angled Artin groups.

Surveying connections between graph combinatorics and algebraic right-angled Artin groups.

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…

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.

Right-angled Artin groups are classified based on 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.

The study examines subgroups of RACGs and RAAGs, focusing on their RAAG properties.

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.

The study broadens the concept of cyclic polytopes to Veronese polytopes.

Explicitly generates right-angled Artin subgroups from mapping classes.

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…

To any prime alternating link, we associate a collection of hyperbolic right-angled ideal polyhedra by relating geometric, topological and combinatorial methods to decompose the link complement. The sum of the hyperbolic volumes of these polyhedra is a new geometric link invariant, which we call the right-angled volume…

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 …

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 $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_Γ$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometr…

Study on finiteness property of right-angled Artin groups actions on extension graphs.

Improved volume estimates for right-angled polyhedra in hyperbolic space.

As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space $\mathbb{H}^n$ has at least one cusp for $n\geq 5$. 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.

New right-angled Artin subgroups found in Artin groups.

Proves involutions on Right-angled Coxeter groups without fixed points.

The paper studies deformation spaces of Coxeter truncation polytopes.

Neural networks 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…

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.

Classifies divergence and thickness in right-angled Coxeter groups.