Related papers: Geodesics with one self-intersection, and other st…

We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichm\"uller space of S. We prove that the composition of these…

For a hyperbolic surface S of finite type we consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of \pi in A(S).

We extend results of Bhagwat and Rajan on a strong multiplicity one property for length spectrum to hyperbolic manifolds with cusps, showing that for two even dimensional hyperbolic manifolds of finite volume, if all but finitely many…

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as…

We prove that for any reversible Finsler metric on S2, the number of prime closed geodesics grows quadratically with respect to length. The main tools are an improvement on Franks' theorem about the number of periodic points of…

Let $S$ be a Riemann surface with a puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, $(a, f^m(a))$ fills $S$ for…

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite…

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T_1(S) x T_1(S), we obtain a local asymptotic equidistribution result for long closed geodesics on…

Let $\Sigma$ be a closed hyperbolic surface. We study, for fixed $g$, the asymptotics of the number of those periodic geodesics in $\Sigma$ having at most length $L$ and which can be written as the product of $g$ commutators. The basic idea…

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…

Let $\mathcal T$ be the Teichm\"{u}ller space of marked genus $g$, $n$ punctured Riemann surfaces with its bordification $\Tbar$ the {\em augmented Teichm\"{u}ller space} of marked Riemann surfaces with nodes, \cite{Abdegn, Bersdeg}.…

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we show that a geodesically complete hyperbolic surface is made up of its convex…

In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter isometry group. As an application of this result, we provide a new proof of the fact that every…

For an infinite type surface $\Sigma$, we consider the space of (marked) convex hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing…

We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is…

In an infinite-dimensional Teichm\"uller space, it is known that the geodesic connecting two points can be unique or not. In this paper, we study the situation on the geodesic in the universal asymptotic Teichm\"uller space $AT(\Delta)$. We…

Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its…