Geometric and Functional Analysis

, Volume 17, Issue 4, pp 1201–1236

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



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 (Fk) 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 


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Copyright information

© Birkhaeuser 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Section de mathématiquesUniversité de GenèveGenèveSwitzerland

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