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

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 $CV(F_k)$ 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.

An updated version

References in corpus (2)

• Growth in free groups (and other stories)
• Minimal volume entropy on graphs

Cited by in corpus (13)

• Geometric Intersection Number and analogues of the Curve Complex for free groups
• Intersection form, laminations and currents on free groups
• The isometry group of Outer Space
• Ping-pong and Outer space
• Stabilizers of $\mathbb{R}$-trees with free isometric actions of $F_N$
• Subset currents on free groups
• The actions of $Out(F_k)$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility
• Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space
• Rigidity of Almost-Isometric Universal Covers
• An intersection functional on the space of subset currents on a free group
• On the absence of McShane-type identities for the outer space
• Growth of balls in the universal cover of surfaces and graphs
• Random length-spectrum rigidity for free groups