Geometric and Functional Analysis

, Volume 17, Issue 4, pp 1201–1236

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

Article

DOI: 10.1007/s00039-007-0621-z

Cite this article as:
Kapovich, I. & Nagnibeda, T. GAFA Geom. funct. anal. (2007) 17: 1201. doi:10.1007/s00039-007-0621-z

Abstract.

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 groupsmetric graphsPatterson–Sullivan measuresgeodesic currentsvolume entropy

AMS Mathematics Subject Classification:

Primary 20F65Secondary 05C, 37A, 37E, 57M

Copyright information

© Birkhaeuser 2007

Authors and Affiliations

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