arXiv research

Study homology growth in nonpositive curvature spaces, finding examples of torsion.

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…

We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of dia…

Homology growth of specific mapping tori vanishes for certain groups.

We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Fi…

Study on triharmonic curves in Sol space with constant curvature and torsion.

Deep learning predicts protein structures accurately.

Extends growth properties of hyperbolic groups to their extensions.

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…

The paper computes torsion invariants for groups acting on complexes.

We classify closed, topological spin$^+$ 4-manifolds with fundamental group $π$ of cohomological dimension $\leq 3$ (up to s-cobordism), after stabilization by connected sum with at most $b_3(π)$ copies of $S^2\times S^2$. In general we must also assume that $π$ also satisfies certain K-theory and assembly map conditio…

Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…

Study the module structure of homology of Artin kernels.

Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.

Right-angled Artin groups have unique quasi-isometry classes when measure equivalent.

Establishes necessary conditions for cylindrical curves using curvature and torsion.

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" grap…

Let P be the right-angled dodecahedron or 120-cell in hyperbolic space, and let W be the group generated by reflections across codimension-one faces of P. We prove that if Gamma is a torsion-free subgroup of minimal index in W, then the corresponding hyperbolic manifold H^n/Gamma is determined up to homeomorphism by Ga…

Let G be a real semisimple Lie group with no compact factors and finite centre, and let $Λ$ be a lattice in G. Suppose that there exists a homomorphism from $Λ$ to the outer automorphism group of a right-angled Artin group $A_Γ$ with infinite image. We give an upper bound to the real rank of G that is determined by the…

Formulae quantify gaps in geodesic quadrilaterals on manifolds.

The paper examines how edge subdivisions affect the vanishing of $L^2$-homology in Coxeter groups.

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…

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…

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and run…

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy t…

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geometric values such as volumes …

The study characterizes loxodromes on specific rotational surfaces in 3D space.

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite of the non-commutativity between functions and (gen…

Defines Kahler angle for a broader context.

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…

Study angle structures on pseudo 3-manifolds, proving existence for some cases.

In the first part of this paper we prove that the mapping class subgroups generated by the $D$-th powers of Dehn twists (with $D\geq 2$) along a sparse collection of simple closed curves on an orientable surface are right angled Artin groups. The second part is devoted to power quotients, i.e. quotients by the normal s…

Introduces a new geometry based on difference angles, showing unique properties.

The paper examines rigidity in geometric actions of Coxeter groups on Croke-Kleiner spaces.

Stiefel-Whitney classes of moment-angle manifolds are trivial.

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…

Study graph products of groups, classifying them up to measure equivalence and rigidity.

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

This note generalizes the visual angle to convex sets in 3D space.

Study proves existence of weak mean curvature flow with contact angle.

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

This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.

Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.

Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubins…

We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…

Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variet…