ﻻ يوجد ملخص باللغة العربية
Let $gamma_0$ be a curve on a surface $Sigma$ of genus $g$ and with $r$ boundary components and let $pi_1(Sigma)curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves $gamma$ of type $gamma_0$ with translation length at most $L$ on $X$. For example, as an application, we derive that for any finite generating set $S$ of $pi_1(Sigma)$ the limit $$lim_{Ltoinfty}frac 1{L^{6g-6+2r}}{gammatext{ of type }gamma_0text{ with }Stext{-translation length}le L}$$ exists and is positive. The main new technical tool is that the function which associates to each curve its stable length with respect to the action on $X$ extends to a (unique) continuous and homogenous function on the space of currents. We prove that this is indeed the case for any action of a torsion free hyperbolic group.
The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functi
We show that Mirzakhanis curve counting theorem also holds if we replace surfaces by orbifolds.
This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.
Let $Sigma$ be a hyperbolic surface. We study the set of curves on $Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $gamma_0$. For example, in the particular case that $Sigma$ is a once-punctured to
We prove and explore a family of identities relating lengths of curves and orthogeodesics of hyperbolic surfaces. These identities hold over a large space of metrics including ones with hyperbolic cone points, and in particular, show how to extend a