# Currents on free groups

@article{Kapovich2006CurrentsOF, title={Currents on free groups}, author={Ilya Kapovich}, journal={arXiv: Group Theory}, year={2006}, pages={149-176} }

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.

## 75 Citations

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

- 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

- 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

- Mathematics
- 2006

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

- Mathematics
- 2020

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

- MathematicsAlgebraic & Geometric Topology
- 2019

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

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- Mathematics
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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

Geodesic currents on negatively curved groups

- Mathematics
- 1991

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

- MathematicsErgodic Theory and Dynamical Systems
- 1997

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

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- 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

- MathematicsErgodic Theory and Dynamical Systems
- 1998

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

- MathematicsInt. J. Algebra Comput.
- 2005

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…