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

Kapovich, Ilya
Published in Geometric and Functional Analysis. 2007, vol. 17, no. 4, p. 1201-1236
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 Free groupsMetric graphsPatterson–Sullivan measuresGeodesic currentsVolume entropy
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KAPOVICH, Ilya, SMIRNOVA-NAGNIBEDA, Tatiana. The Patterson-Sullivan embedding and minimal volume entropy for outer space. In: Geometric and Functional Analysis, 2007, vol. 17, n° 4, p. 1201-1236. doi: 10.1007/s00039-007-0621-z https://archive-ouverte.unige.ch/unige:6537

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Deposited on : 2010-05-07

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