arXiv research

Positive Thompson links are proven for oriented subgroup elements.

Positive Thompson links are arborescent tangles.

The study finds non-trivial elements in moduli spaces of curvature metrics.

New method counts link components from Thompson group elements.

It is well known that if the dimension of the Sasaki cone is greater than one, then all Sasakian structures are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that the dimension of the Sasaki cone is greater than one, there are three possibiliti…

Proves existence of infinite order elements in curvature spaces.

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with t…

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface $Σ_g$, which are rotations of $Σ_g$ through $2π/p$, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.

The study examines conjugation curvature in a specific group, finding elements with various curvatures.

The paper constructs hyperbolic elements in multiple spaces.

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical …

Characterizes periodic elements in Artin-Tits groups via stability conditions.

Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.

A new algorithm FastGM speeds up generating Gumbel-Max variables.

The paper proposes a notion of volume element for Finsler spaces with metrics of Lorentzian signature, equipped with a time orientation. This notion is based on a slight modification of the idea of Holmes-Thompson volume element working for positive definite Finsler metrics and can be used in field-theoretical applicat…

Let Gamma be a group generated by two positive multi-twists. We give some sufficient conditions for Gamma to be free or have no `unexpectedly reducible' elements. For a group Gamma generated by two Dehn twists, we classify the elements in Gamma which are multi-twists. As a consequence we are able to list all the lanter…

Proves mapping class group generated by two torsion elements for certain surfaces.

We construct non-trivial elements of order 2 in the homotopy groups $π_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Mor…

Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.

A positive integer $m$ will be called a {\it finitistic order} for an element $γ$ of a group $Γ$ if there exist a finite group $G$ and a homomorphism $h:Γ\to G$ such that $h(γ)$ has order $m$ in $G$. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian gr…

Whether you trade futures for yourself or a hedge fund, your strategy is counted. Long and short position limits make the number of unique strategies finite. Formulas of the numbers of strategies, transactions, do nothing actions are derived. A discrete distribution of actions, corresponding probability mass, cumulativ…

The well-known fact that any genus $g$ symplectic Lefschetz fibration $X^{4}\to S^{2}$ is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manif…

Study random walks on CAT(0) spaces with contracting elements, proving limit laws.

Let $G$ be a Garside group with Garside element $Δ$. An element $g$ in $G$ is said to be \emph{periodic} if some power of $g$ lies in the cyclic group generated by $Δ$. This paper shows the following. (i) The periodicity of an element does not depend on the choice of a particular Garside structure if and only if the ce…

New method detects foliation enlargeability.

We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoret…

A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1,1)-position. Afte…

Let $Γ$ be a finite index subgroup of the mapping class group $MCG(Σ)$ of a closed orientable surface $Σ$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\inΓ$ has positive stable commutator length. In addition, we show that in these situations th…

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero l…

In this note, we prove a lower bound for the positive kinkiness of a closed braid which we then use to derive an estimate for the positive kinkiness of a link in terms of its Seifert system. As an application, we show that certain pretzel knots cannot be unknotted using only positive crossing changes. We also describe …

We consider $n$-dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation ("rotating motions"). In the Study quadric model of Euclidean displacements these motions correspond to quadrilateral nets with edges contained in the Study quadric ("rotation nets"). The main focus of ou…

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(o, n)$ (resp. $\mathcal C'(o, n)$) the set of (resp. primitive) conjugacy classes of pointed length at most $n$ for a basep…

Study on braids' invariant growth and counting functions.

Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of …

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length o…

Given a set $V$ of $n$ objects, an online ranking system outputs at each time step a full ranking of the set, observes a feedback of some form and suffers a loss. We study the setting in which the (adversarial) feedback is an element in $V$, and the loss is the position (0th, 1st, 2nd...) of the item in the outputted r…

New findings on cusped Borel Anosov representations and their properties.

Natural language processing improves COVID-19 hospitalization identification.

The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.

Let $G$ be a Garside group with Garside element $Δ$, and let $Δ^m$ be the minimal positive central power of $Δ$. An element $g\in G$ is said to be 'periodic' if some power of it is a power of $Δ$. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an…

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should…

Floer cohomology is computed for certain elements of the mapping class group of a surface $Σ$ of genus $g>1$ which are compositions of positive and negative dehn twists along some loops in $Σ$. The computations cover a certain class of pseudo-Anasov maps.

Well-defined formal definitions for sentiment and opinion are extended to incorporate the necessary elements to provide a formal quantitative definition of reputation. This definition takes the form of a time-based index, in which each element is a function of a collection of opinions mined during a given time period. …

We show that for a closed Riemannian manifold the quotient of the group of projective transformations by the group of isometries contains at most two elements unless the metric has constant positive sectional curvature or every projective transformation is an affine transformation.

New results show metrics with positive scalar curvature can cancel on certain 4-manifolds.

The language of maximal lexicographic representatives of elements in the positive braid monoid $A_n$ with $n$ generators is a regular language. We describe with great detail the smallest Finite State Automaton accepting such language, and study the proportion of elements of length $k$ whose maximal lexicographic repres…

Proves infinite bordism groups for certain manifolds with positive scalar curvature.