Counting curves, and the stable length of currents

  • Viveka Erlandsson

    University of Bristol, UK
  • Hugo Parlier

    University of Luxembourg, Luxembourg
  • Juan Souto

    Université de Rennes 1, France
Counting curves, and the stable length of currents cover
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Let be a curve on a surface of genus and with boundary components and let be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves of type with translation length at most on . For example, as an application, we derive that for any finite generating set , of the limit

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 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.

Cite this article

Viveka Erlandsson, Hugo Parlier, Juan Souto, Counting curves, and the stable length of currents. J. Eur. Math. Soc. 22 (2020), no. 6, pp. 1675–1702

DOI 10.4171/JEMS/953