arXiv research

For a surface group, a new bound is given for conjugator length function.

In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger--Mozes-ty…

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

The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.

Smoothly conjugate Anosov flows on 3D manifolds are actually smoothly conjugate.

Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial differential equations controlling the functions of three variables that admit a c…

Upper bound on Stiefel manifold's injectivity radius found.

Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which a…

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2π$. With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'

We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conj…

The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…

In this paper we explore relationships between divergence and thick groups, and with the same techniques we estimate lengths of shortest conjugators. We produce examples, for every positive integer n, of CAT(0) groups which are thick of order n and with polynomial divergence of order n+1, both these phenomena are new. …

Study geodesics on Grushin spaces, proving upper bounds on conjugate times.

Study finds conjugate points in geodesics of Kolmogorov flows on torus.

This is essentially a note on Section 7 of Perelman's first paper on Ricci flow. We list some basic properties of the index form for Perelman's $\mathcal{L}$-length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse's index theorem for Perelman's $\mathcal{L}$-length holds…

Random groups prove length constraints on product of conjugates.

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously known results that each solvable subgroup of the fundamental group is a Bieberbac…

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union…

New lattice extensions of Schottky groups in hyperbolic space.

We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.

What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by t…

The paper generalizes rigidity results for contact Anosov flows with bunching assumption.

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly det…

In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those with support in balls and conjugation-generated norm.

New displacement technique vanishes bounded cohomology in all degrees.

The scattering data of a Riemannian manifold with boundary record the incoming and outgoing directions of each geodesic passing through. We show that the scattering data of a generic Riemannian surface with no trapped geodesics and no conjugate points determine the lengths of geodesics. Counterexamples exists when trap…

RNA structures show that a significant portion of bases do not form hydrogen bonds.

We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.

We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infin…

We describe the family of minimal graphs on strips with boundary values $\pm\infty$ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in $\mathbb{R}^3$. We can obtain as limits of such graphs the helicoid, all the doubly periodic Scherk minimal surfac…

We prove three facts about intrinsic geometry of surfaces in a normed (Minkowski) space. When put together, these facts demonstrate a rather intriguing picture. We show that (1) geodesics on saddle surfaces (in a space of any dimension) behave as they are expected to: they have no conjugate points and thus minimize len…

In this paper, we prove that, for any integer $n\ge 2,$ there exists an $ε_{n} \ge 0$ so that if $M$ is an n-dimensional complete manifold with sectional curvature $K_{M}\ge 1$ and if $M$ has conjugate radius bigger than $\fracπ{2}$ and contains a geodesic loop of length $2(π-ε_{n}),$ then $M$ is diffeomorphic to the…

This paper studies spectral properties of spheres with one equator.

Global rigidity theorem for metrics on Σ×S1 without conjugate points.

The paper examines boundedness of diffeomorphism groups of manifold pairs, focusing on the circle case.

Metrics are isometric for certain Anosov magnetic systems.

Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is stated which implies uniqueness and stability for this problem. If the conformal c…

New findings on translation lengths in Teichmüller and curve graphs for pseudo-Anosovs.

Let $σ$ be the scattering relation on a compact Riemannian manifold $M$ with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let $\ell$ be the length of that geodesic ray. We study the que…

We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal conjugation action …

The fundamental group of the complement of a plane curve is a very important topological invariant. In particular, it is interesting to find out whether this group is determined by the combinatorics of the curve or not, and whether it is a direct sum of free groups and a free abelian group, or it has a conjugation-free…

We prove that the topology, smooth structure, and metric of a compact Lorentzian manifold with boundary is uniquely determined by data at the boundary. The data consists of the lengths and directions of future-directed once-broken geodesics connecting points on the boundary, which are first timelike and then lightlike.…

Two Anosov metrics with same boundary distance are isometric.

For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants {τ_n(M; η)}. Roughly speaking, {τ_n(M; η)} is the Reidemeister torsion of M with respect to the representation given by the composition of the lift of the holonomy representation defined by η, and t…

The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same trave…

Consider a compact Riemannian manifold with boundary. Assume all maximally extended geodesics intersect the boundary at both ends. Then to each maximal geodesic segment one can form a triple consisting of the initial and final vectors of the segment and the length of the segment. The collection of all such triples comp…

A Riemannian manifold is said to be uniformly secure if there is a finite number $s$ such that all geodesics connecting an arbitrary pair of points in the manifold can be blocked by $s$ point obstacles. We prove that the number of geodesics with length $\leq T$ between every pair of points in a uniformly secure manifol…

We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motiv…