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Register a new userCounting curves, and the stable length of currents
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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.
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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 functions to provide a proof of a folklore theorem which states that pseudo-Anosov homeomorphisms of closed hyperbolic surfaces act on the space of projective geodesic currents with uniform north-south dynamics.
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 torus, we prove that the cardinality of the set of curves of type $gamma_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.
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 result of the first author to surfaces with cusps. One of the main ingredients in the approach is a partition of the set of orthogeodesics into sets depending on their dynamical behavior, which can be understood geometrically by relating them to geodesics on orbifold surfaces. These orbifold surfaces turn out to be exactly on the boundary of the space in which the underlying identity holds.
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