# Currents on free groups

@article{Kapovich2006CurrentsOF,
title={Currents on free groups},
author={Ilya Kapovich},
journal={arXiv: Group Theory},
year={2006},
pages={149-176}
}
• Ilya Kapovich
• Published 7 December 2004
• Mathematics
• arXiv: Group Theory
We study the properties of geodesic currents on free groups, particularly the "intersection form" that is similar to Bonahon's notion of the intersection number between geodesic currents on hyperbolic surfaces.
EQUIVALENCE IN FREE GROUPS
• Mathematics
• 2006
Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when
TRANSLATION EQUIVALENCE IN FREE GROUPS
Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when
Translation equivalence in free groups
• Mathematics
• 2004
Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when
Current twisting and nonsingular matrices
• Mathematics
• 2009
We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.
Geodesic currents and length compactness for automorphisms of free groups
We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping the length of the uniform current bounded is compact (up to conjugation). This
GEODESIC CURRENTS AND LENGTH COMPACTNESS FOR AUTOMORPHISMS OF FREE GROUPS
We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping the length of the uniform current bounded is compact (up to conjugations.) This
A hyperbolic Out(Fn)-complex
• Mathematics
• 2010
For any finite collection fi of fully irreducible automorphisms of the free group Fn we construct a connected i-hyperbolic Out.Fn/-complex in which each fi has positive translation length.
Connectivity of the Gromov Boundary of the Free Factor Complex
• Mathematics
• 2021
We show that in large enough rank, the Gromov boundary of the free factor complex is path connected and locally path connected.
Hyperbolic extensions of free groups from atoroidal ping-pong
We prove that all atoroidal automorphisms of $Out(F_N)$ act on the space of projectivized geodesic currents with generalized north-south dynamics. As an application, we produce new examples of non
$\R$-trees and laminations for free groups III: Currents and dual $\R$-tree metrics
• Mathematics
• 2008
This is the third of a series of three articles where we introduce laminations for the free-groups. We explore here the link between currents and laminations and prove that the situation is more

## References

SHOWING 1-10 OF 30 REFERENCES
Ergodic Properties of Function Groups
Let Γ ⊂PSL(2,ℂ) be a torsion free function group. We discuss some aspects of the asymptotic relation between the geometry of the hyperbolic manifold H3/Γ and the geometry of the boundary of its
Translation equivalence in free groups
• Mathematics
• 2004
Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when
Geodesic currents on negatively curved groups
A negatively curved or hyperbolic group, as introduced by M. Gromov, is a finitely generated group whose Cayley graphs asymptotically behave at infinity like a tree. Considering the action of a
Non-uniquely ergodic foliations of thin type
We construct a minimal foliation of thin type which is not uniquely ergodic. The notion of thin type relates to Rips' classification of foliations on 2-complexes.
The symmetries of outer space
• Mathematics
• 2001
For n ≥ 3, the natural map Out(Fn)→ Aut(Kn) from the outer automorphism group of the free group of rank n to the group of simplicial automorphisms of the spine of outer space is an isomorphism. §
Equivalence of boundary measures on covering trees of finite graphs
• R. Lyons
• Mathematics
Ergodic Theory and Dynamical Systems
• 1994
Abstract Let T be the universal covering tree of a finite graph, G. By analogy with an open problem concerning negatively curved covering manifolds, Kaimanovich asked when two of the three natural
The topology at infinity of Out(Fn)
• Mathematics
• 2000
Abstract.We construct a bordification of Outer Space analogous to the Borel-Serre bordification of symmetric spaces. As an application, we show that Out(Fn) is (2n-5)-connected at infinity and that
Hausdorff dimension of the harmonic measure on trees
For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$
The Frequency Space of a Free Group
It is proved that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, is the intersection ofℚ and a closed subinterval of ℝ with rational endpoints.
Moduli of graphs and automorphisms of free groups
• Mathematics
• 1986
This paper represents the beginning of an a t tempt to transfer, to the study of outer au tomorphisms of free groups, the powerful geometric techniques that were invented by Thurs ton to study