arXiv research

Solves double coset problem for braid group H_n.

The study of double coset growth in specific groups confirms a conjecture about generic 3-manifolds.

Paper explores link and plat presentations, showing equivalence under bridge isotopy.

New methods convert complex link presentations to simpler, recognizable forms.

Let M = H^3 / Γbe a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of Γand g is in Γ, then the double coset HgK is separable in Γ. As a consequence we prove that if M is a closed, orientable, Haken 3-manifold and the fundamental group of every hyperbolic piece of the torus decompo…

Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n_0 \leq …

The paper classifies differentiable structures on a line with two origins.

In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.

We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain…

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is…

The paper extends group constructions to coset geometries, creating new ways to combine geometries.

Develops intrinsic curved cosets for Cartan geometries.

Mapping class group subgroups yield quasi-isometric curve complex.

Study shows stable graphs in Heisenberg group are essentially planes.

3-manifolds have covers with infinitely many ideal triangulations.

We prove that the Cayley graph and the coset geometry of the von Dyck group $D(a,b,c)$ are linked by a vertex-to-edge duality.

As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the r…

We use geometric methods to show that given any $3$-manifold $M$, and $g$ a sufficiently large integer, the mapping class group $\mathrm{Mod}(Σ_{g,1})$ contains a coset of an abelian subgroup of rank $\lfloor \frac{g}{2}\rfloor,$ consisting of pseudo-Anosov monodromies of open-book decompositions in $M.$ We prove a sim…

Neural program embedding can be helpful in analyzing large software, a task that is challenging for traditional logic-based program analyses due to their limited scalability. A key focus of recent machine-learning advances in this area is on modeling program semantics instead of just syntax. Unfortunately evaluating su…

It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…

In this paper, we formulate a construction of ideal coset invariants for surface-links in $4$-space using invariants for knots and links in $3$-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an invariant for surface-links called the Kauffman bracket ideal coset invariant of sur…

Proves cosets of certain subgroups in hyperbolic 3-manifold groups are conjugacy distinguished.

J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…

In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…

The Cayley graph of quandles reveals structural properties and is studied for various classes.

A compact Riemannian homogeneous space $G/H$, with a bi--invariant orthogonal decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in $T_{eH}(G/H)$ spanned by a linearly independent commuting pair in $\mathfrak{…

New metrics and coordinates for barcode space using group theory.

Paper classifies fibers of fat Riemannian submersions with non-negative curvature.

We provide an algorithm to solve the word problem in all fundamental groups of closed 3-manifolds; in particular, we show that these groups are autostackable. This provides a common framework for a solution to the word problem in any closed 3-manifold group using finite state automata. We also introduce the notion of a…

In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that t…

Strong bolicity helps prove Baum-Connes conjecture for certain hyperbolic groups.

Complex captures group properties, invariant under quasi-isometry.

We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of $\mathbb{H}$. Moreover two examples are given…

We review the basic elements of the geometrical formalism for description of gauge fields and the theory of invariant connections, and their applications to the coset space dimensional reduction of Yang-Mills theories. We also discuss the problem of classification of principal fibre bundles, which is important for the …

For i = 1,2, let Gamma_i be a lattice in a simply connected, solvable Lie group G_i, and let X_i be a connected Lie subgroup of G_i. The double cosets Gamma_igX_i provide a foliation F_i of the homogeneous space Gamma_i\G_i. Let f be a continuous map from Gamma_1\G_1 to Gamma_2\G_2 whose restriction to each leaf of F_1…

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact …

Defines a strict order on plat presentation classes for links.

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry…

The paper consider the symmetric of Finsler spaces. We give some conditions about globally symmetric Finsler spaces. Then we prove that these spaces can be written as a coset space of Lie group with an invariant Finsler metric. Finally, we prove that such a space must be Berwaldian

For every finite graph $Γ$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_Γ$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_Γ)$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotop…

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli opera…

New surfaces in 3D manifolds are found that cannot be smoothly deformed into each other.

Kapovich and Nagnibeda introduced the space $\mathcal{S} {\rm Curr}(F_N)$ of subset currents on a free group $F_N$ of rank $N\geq 2$, which can be thought of as a measure-theoretic completion of the set of all conjugacy classes of finitely generated subgroups of $F_N$. We define a product $\mathcal{N} (H,K)$ of two fin…

In this paper we give a realization of some symmetric space G/K as a closed submanifold P of G. We also give several equivalent representations of the submanifold P. Some properties of the set gK\cap P are also discussed, where gK is a coset space in G.

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal ho…

Characterizes critical points in convex double and triple bubbles.

Let S be a closed, oriented surface of genus at least 2, and consider the extension 1 -> pi_1 S -> MCG(S,p) -> MCG(S) -> 1, where MCG(S) is the mapping class group of S, and MCG(S,p) is the mapping class group of S punctured at p. We prove that any quasi-isometry of MCG(S,p) which coarsely respects the cosets of the no…

In 86, Ranjan questioned whether a submersion $π: G\to B$ from a compact simple Lie group with bi-invariant metric is a coset foliation or not, provided the submersion is Riemannian with totally geodesic fibers. Here we answer this question affirmatively, even when the submersion is defined only in an open subset of $G…