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

*(English)*Zbl 1135.20031Given a free group \(F\) of rank \(k\geq 2\), its ‘Culler-Vogtmann’ space \(CV(F)\), also called ‘outer space’, is the set of equivalence classes of free discrete and minimal isometric actions of \(F\) on \(\mathbb{R}\)-trees for which the quotient metric has volume one, and where equivalence is defined by the existence of an \(F\)-equivariant isometry between the two trees.

The authors consider the space of geodesic currents on the free group, that is, positive \(F\)-invariant Radon measures on the space \(\partial^2F=\{(x,y)\mid x,y\in F\), \(x\not=y\}\). This theory parallels Bonahon’s geodesic currents on surface groups. Bonahon used geodesic currents in the study of Teichmüller space, and the authors in the paper under review use geodesic currents to study outer space, which plays for outer automorphisms of free groups the role that Teichmüller space plays for the mapping class group.

The authors, in analogy with Bonahon’s work, consider Patterson-Sullivan currents for actions of free groups on trees. They define a map \(\tau\colon CV(F)\to\mathbb{P}\text{Curr}(F)\) that assigns to a point in \(CV(F)\), represented by an action of \(F\) on a tree, the projective class of the Patterson-Sullivan current corresponding to this action. The main result they obtain is that the Patterson-Sullivan map \(\tau\) is a continuous embedding. They also define a “Hausdorff dimension map” \(h\colon CV(F)\to\mathbb{R}\) and they prove it is continuous and that the restriction of this map to any open simplex in \(CV(F)\) is real-analytic.

The continuous embedding \(\tau\) gives a new compactification of outer space.

The authors also prove that for every \(k\geq 2\), 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.

The authors consider the space of geodesic currents on the free group, that is, positive \(F\)-invariant Radon measures on the space \(\partial^2F=\{(x,y)\mid x,y\in F\), \(x\not=y\}\). This theory parallels Bonahon’s geodesic currents on surface groups. Bonahon used geodesic currents in the study of Teichmüller space, and the authors in the paper under review use geodesic currents to study outer space, which plays for outer automorphisms of free groups the role that Teichmüller space plays for the mapping class group.

The authors, in analogy with Bonahon’s work, consider Patterson-Sullivan currents for actions of free groups on trees. They define a map \(\tau\colon CV(F)\to\mathbb{P}\text{Curr}(F)\) that assigns to a point in \(CV(F)\), represented by an action of \(F\) on a tree, the projective class of the Patterson-Sullivan current corresponding to this action. The main result they obtain is that the Patterson-Sullivan map \(\tau\) is a continuous embedding. They also define a “Hausdorff dimension map” \(h\colon CV(F)\to\mathbb{R}\) and they prove it is continuous and that the restriction of this map to any open simplex in \(CV(F)\) is real-analytic.

The continuous embedding \(\tau\) gives a new compactification of outer space.

The authors also prove that for every \(k\geq 2\), 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.

Reviewer: Athanase Papadopoulos (Strasbourg)

##### MSC:

20F65 | Geometric group theory |

05C12 | Distance in graphs |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |

37E25 | Dynamical systems involving maps of trees and graphs |

57M07 | Topological methods in group theory |

20E08 | Groups acting on trees |