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Register a new userHyperbolicity of the complex of free factors
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We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.
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We show that in large enough rank, the Gromov boundary of the free factor complex is path connected and locally path connected.
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.
We study the large scale geometry of the relative free splitting complex and the relative complex of free factor systems of the rank $n$ free group $F_n$, relative to the choice of a free factor system of $F_n$, proving that these complexes are hyperbolic. Furthermore we present the proof in a general context, obtaining hyperbolicity of the relative free splitting complex and of the relative complex of free factor systems of a general group $Gamma$, relative to the choice of a free factor system of $Gamma$. The proof yields information about coarsely transitive families of quasigeodesics in each of these complexes, expressed in terms of fold paths of free splittings.
This paper is devoted to the computation of the space $H_b^2(Gamma,H;mathbb{R})$, where $Gamma$ is a free group of finite rank $ngeq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $Gamma$ if and only if $H_b^2(Gamma,H;mathbb{R})$ is not trivial, and furthermore, if and only if there is an isometric embedding $oplus_infty^nmathcal{D}(mathbb{Z})hookrightarrow H_b^2(Gamma,H;mathbb{R})$, where $mathcal{D}(mathbb{Z})$ is the space of bounded alternating functions on $mathbb{Z}$ equipped with the defect norm.
For a fully irreducible automorphism phi of the free group F_k we compute the asymptotics of the intersection number n mapsto i(T,Tphi^n) for trees T,T in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and Tphi^n for n large.
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