Counting curves, and the stable length of currents

Viveka Erlandsson, Hugo Parlier, Juan Souto

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

Let γ0 be a curve on a surface Σ of genus g and with r boundary components and let π1(Σ)↷X be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves γ of type γ0 with translation length at most L on X. For example, as an application, we derive that for any finite generating set S of π1(Σ) the limit limL→∞1L6g−6+2r{γ of type γ0 with S-translation length≤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.
Original languageEnglish
Pages (from-to)1675–1702
Number of pages28
JournalJournal of the European Mathematical Society
Volume22
Issue number6
DOIs
Publication statusPublished - 17 Feb 2020

Keywords

  • Growth of geodesics on surfaces
  • currents
  • hyperbolic groups

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