Related papers: Equal angles of intersecting geodesics for every h…

We provide a new and simple system of equations for the normal sub-Riemannian geodesics. These use a partial connection that we show is canonically available, given a choice of complement to the distribution. We also describe conditions…

We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding…

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the…

On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs…

Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial \Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper $\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated…

In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the…

We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order.

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of…

We prove that if the $m$-th homotopy group for $m \geq 2$ of a closed manifold has non-trivial invariants or coinvariants under the action of the fundamental group, then there exist infinitely many geometrically distinct closed geodesics…

In this paper we study the equilibrium measures of geodesic flows of closed manifolds without conjugate points which have a visibility universal covering. Specifically, the uniqueness problem for Bowen potentials which are constants on some…

We show that both Teichmuller space (with the Teichmuller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic…

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length…

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this problem to hyperbolic 3-manifolds admitting tunnel systems. We show…

We investigate various homotopy invariant formulations of commutative algebra in the context of rational homotopy theory. The main subject is the complete intersection condition, where we show that a growth condition implies a structure…

We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we…

For a finite group $H$ and connected topological spaces $X$ and $Y$ such that $X$ is endowed with a free left $H$-action $\tau$, we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the…

In this note we prove that the symmetry of the quasi-isometry relation implies the axiom of choice, even when the relation is restricted to geodesic hyperbolic spaces. We show that this result is sharp by demonstrating that symmetry of…

We prove a generalization of the fellow traveller property for a certain type of quasi-geodesics and use it to present three equivalent geometric formulations of the bounded reduction property and prove that it is equivalent to preservation…

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and…