Related papers: Equal angles of intersecting geodesics for every h…
A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of…
In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the…
Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on the surface, the length of the geodesic in one class is equal to the…
We study closed geodesics on hyperbolic surfaces, and give bounds for their angles of intersection and self-intersection, and for the sides of the polygons that they form, depending only on the lengths of the geodesics
In this note we develop a tool box of non-Euclidean plane geometry methods that yield a constructive way to define in terms of closed geodesics the Goldman bracket on deformation classes of closed, directed curves. We use this construction…
We provide some explicit algebraic criteria in terms of the Goldman bracket to decide whether two free homotopy classes of loops on an oriented surface admit disjoint representatives. We extend Kabiraj's method using the hyperbolic geometry…
We show that a geodesic metric space is hyperbolic in the sense of Gromov if and only if intersections of balls have bounded eccentricity. In particular, $\R$-trees are characterized among geodesic metric spaces by the property that the…
Let X be a complete hyperbolic surface of finite area. We establish that the intersection points of closed geodesics with length <T are equidistributed on X as T goes to infinity.
We show that a homotopy equivalence between two non-compact orientable surfaces is homotopic to a homeomorphism if and only if it preserves the Goldman bracket, provided our surfaces are neither the plane nor the punctured plane.
Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic…
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…
In an orientable surface with boundary, free homotopy classes of curves on surfaces are in one to one correspondence with cyclic reduced words in a set of standard generators of the fundamental group. The combinatorial length of a class is…
We prove that, if a closed geodesic $\Gamma$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $\Gamma$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a…
We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these…
We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups…
We study the properties of geodesic currents on free groups, particularly the "intersection form" that is similar to Bonahon's notion of the intersection number between geodesic currents on hyperbolic surfaces.
By a geodesic subspace of a metric space $X$ we mean a subset $A$ of $X$ such that any two points in $A$ can be connected by a geodesic in $A$. It is easy to check that a geodesic metric space $X$ is an $\mathbb{R}$-tree (that is, a…
We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of…
We show that the center of the Goldman algebra associated to a closed oriented hyperbolic surface is trivial. For a hyperbolic surface of finite type with nonempty boundary, the center consists of closed curves which are homotopic to…
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…