We consider the reducibility problem of cocycles by isometries Gromov hyperbolic metric spaces in Livsic setting. show that provided boundary cocycle (that acts on a compact space) is reducible suitable H\"older class, then original an unbounded also reducible.

The hidden metric space behind complex network topologies is a fervid topic in current network science and the hyperbolic space is one of the most studied, because it seems associated to the structural organization of many real complex systems. The PopularitySimilarity-Optimization (PSO) model simulates how random geometric graphs grow in the hyperbolic space, reproducing strong clustering and ...

As we have seen in the previous lecture, Teichmüller space T (S) for a surface of finite type S = Sg,n (genus g and n punctures or boundary components) is a parameter space for various types of structures that can be imposed on the same topological surface S. I’ll adopt the view that T (S) is the space of metrics on S, marked with a set of generators for the fundamental group, up to (measurable...

Recall that a conformal 4-manifold is called self-dual if its Weyl curvature, considered as a bundle valued 2-form, is in the +1 eigenspace of the Hodge star-operator [1]. Due to Schoen’s proof [19] of the Yamabe conjecture it is known that within any conformal class on a compact manifold is a metric whose scalar curvature is constant and the sign of this constant is a conformal invariant. The ...

A. In this article we exhibit the optimal (i.e. largest) constants for the quadratic isoperimetric and the linear filling radius inequality which ensure that a geodesic metric space X is Gromov hyperbolic. Our results show that the Euclidean plane is a borderline case for the isoperimetric inequality. Furthermore, by only requiring the existence of isoperimetric fillings in L∞(X) satisfy...

In this paper we generalize to unbounded convex subsets C of hyperbolic spaces results obtained by W.A. Kirk and R. Esṕınola on approximate fixed points of nonexpansive mappings in product spaces (C×M)∞, where M is a metric space and C is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families (Cu)u∈M of unbounded convex subsets of...

‎The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]‎. ‎In [1]‎, ‎Ungar and Chen showed that the algebra of the group $SL(2,mathbb C)$ naturally leads to the notion of gyrogroups ‎and gyrovector spaces for dealing with the Lorentz group and its ‎underlying hyperbolic geometry‎. ‎They defined the Chen addition and then Chen model of hyperbolic geomet...

Abstract Given a compact orientable surface with finitely many punctures Σ, let S(Σ) be the set of isotopy classes of essential unoriented simple closed curves in Σ. We determine a complete set of relations for a function from S(Σ) to R to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on Σ. As a consequence, the Teichmüller space of hyperbolic metri...

The grid cells discovered in the rodent medial entorhinal cortex have been proposed to provide a metric for Euclidean space, possibly even hardwired in the embryo. Yet, one class of models describing the formation of grid unit selectivity is entirely based on developmental self-organization, and as such it predicts that the metric it expresses should reflect the environment to which the animal ...

We analyze the coarse geometry of the Weil-Petersson metric on Teichmüller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the pants graph. We show that in dimension 3 the pants graph is strongly rela...