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

By Ilya Kapovich, Tatiana Nagnibeda. 2005

51 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/47b76bb3786649a22cb5b16b4d6df696cdc5ee4c

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The actions of ${\mathit{Out}(F_k)}$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility (2006, background): “Other results regarding currents and free group automorphisms can be found in [17, 18 , 19, 20, 21, 22, 18, 24, 27].”