[Paper Review] A hyperbolic Out(F_n)-complex

Mladen Bestvina, Mark Feighn|ArXiv.org|Aug 27, 2008

Geometric and Algebraic Topology23 references16 citations

TL;DR

This paper constructs a connected, $δ$-hyperbolic graph ${\mathcal{X}}$ on which $\mathrm{Out}(F_n)$ acts isometrically, such that any finite collection of fully irreducible automorphisms have positive translation length. The construction adapts Bowditch's method using annulus systems and hyperbolic crossratios, relying on the dynamics of $\mathrm{Out}(F_n)$ on the space of trees and currents, yielding a hyperbolic complex without an intrinsic description akin to the curve complex.

ABSTRACT

For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $δ$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.

Motivation & Objective

• To construct a $\delta$-hyperbolic $\mathrm{Out}(F_n)$-complex where fully irreducible automorphisms act with positive translation length.
• To extend Bowditch's hyperbolicity construction from convergence groups to the $\mathrm{Out}(F_n)$ setting using dynamics on spaces of trees and currents.
• To provide a geometric framework for constructing quasi-homomorphisms on $\mathrm{Out}(F_n)$, as recently announced by Hamenstädt.
• To establish that stabilizers of simplicial trees and proper free factors have bounded orbits in the new complex.
• To demonstrate that the complex is not intrinsic but still useful for studying the geometry of $\mathrm{Out}(F_n)$.

Proposed method

• Adapts Bowditch's construction of a hyperbolic space from a group action on a space with an annulus system satisfying axioms (A1) and (A2).
• Defines a hyperbolic path crossratio on the space of trees and currents using counting of nested annuli.
• Uses the dynamics of $\mathrm{Out}(F_n)$ on the compactified Outer space $\overline{\mathcal{PT}}$ and the space of projectivized currents $\mathcal{M}(F_n)$ to define the annulus system.
• Constructs a quasi-metric $\rho$ on the space of ordered triples of distinct points in $M$, using the crossratio, and proves it defines a hyperbolic path quasi-metric.
• Applies the construction to the action of $\mathrm{Out}(F_n)$ on the space of trees and currents, ensuring hyperbolicity via the crossratio axioms.
• Relies on known results on the dynamics of fully irreducible automorphisms and their north-south dynamics on the boundary of Outer space.

Experimental results

Research questions

• RQ1 Can a $\delta$-hyperbolic $\mathrm{Out}(F_n)$-complex be constructed such that fully irreducible automorphisms have positive translation length?
• RQ2 Does Bowditch's method for constructing hyperbolic spaces from group actions on spaces with annulus systems extend to $\mathrm{Out}(F_n)$?
• RQ3 Can the dynamics of $\mathrm{Out}(F_n)$ on the space of trees and currents be used to define a hyperbolic crossratio?
• RQ4 What are the geometric consequences of such a complex for the study of $\mathrm{Out}(F_n)$, particularly in constructing quasi-homomorphisms?
• RQ5 Is it possible to ensure bounded orbits for stabilizers of simplicial trees and proper free factors in the constructed complex?

Key findings

• For any finite collection of fully irreducible automorphisms $f_1, \dots, f_k$ in $\mathrm{Out}(F_n)$, there exists a connected $\delta$-hyperbolic graph $\mathcal{X}$ with an isometric $\mathrm{Out}(F_n)$-action where each $f_i$ has positive translation length.
• The stabilizer of any simplicial tree in $\overline{\mathcal{PT}}$ has bounded orbits in $\mathcal{X}$, and the same holds for stabilizers of proper free factors.
• The construction relies on a hyperbolic path crossratio derived from an annulus system satisfying axioms (A1) and (A2), ensuring the resulting space is hyperbolic.
• The complex $\mathcal{X}$ is not intrinsic like the curve complex but is still useful for constructing quasi-homomorphisms on $\mathrm{Out}(F_n)$, as shown by Hamenstädt.
• The Cayley graph of $\mathrm{Out}(F_n)$ with respect to a finite generating set contains arbitrarily large balls consisting entirely of fully irreducible automorphisms.
• If $\Gamma$ is an irreducible lattice in a semisimple Lie group of rank $\geq 2$, then any embedding $\Gamma \to \mathrm{Out}(F_n)$ cannot contain any fully irreducible automorphisms.