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

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

2005-04-21

arxiv.org/abs/math/0504445v3

Geom. Funct. Anal. vol. 17 (2007), no. 4, pp. 1201-1236

Mathematics

Group Theory

An updated version

Scientific paper

10.1007/s00039-007-0621-z

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 topological embedding. We also prove that for every $k\ge 2$ 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.

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