Geometric and Functional Analysis

, Volume 17, Issue 4, pp 1201-1236

First online:

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

  • Ilya KapovichAffiliated withDepartment of Mathematics, University of Illinois at Urbana-Champaign Email author 
  • , Tatiana NagnibedaAffiliated withSection de mathématiques, Université de Genève

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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.

Keywords and phrases:

Free groups metric graphs Patterson–Sullivan measures geodesic currents volume entropy

AMS Mathematics Subject Classification:

Primary 20F65 Secondary 05C, 37A, 37E, 57M