The Patterson-Sullivan embedding and minimal volume entropy for outer space

Ilya Kapovich, Tatiana Nagnibeda

Research output: Contribution to journalArticlepeer-review


Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space CV (F k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k - 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs.

Original languageEnglish (US)
Pages (from-to)1201-1236
Number of pages36
JournalGeometric and Functional Analysis
Issue number4
StatePublished - Nov 2007


  • Free groups
  • Geodesic currents
  • Metric graphs
  • Patterson-Sullivan measures
  • Volume entropy

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

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