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Register a new userBoundary amenability of $Out(F_N)$
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We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a finitely generated right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures.
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For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $delta$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
We show that if a f.g. group $G$ has a non-elementary WPD action on a hyperbolic metric space $X$, then the number of $G$-conjugacy classes of $X$-loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $Nge 3$ the number of distinct $Out(F_N)$-conjugacy classes of fully irreducibles $phi$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $loglambda(phi)$ on the order of $R$ grows exponentially in $R$.
$Out(F_n):=Aut(F_n)/Inn(F_n)$ denotes the outer automorphism group of the rank $n$ free group $F_n$. An element $phi$ of $Out(F_n)$ is polynomially growing if the word lengths of conjugacy classes in $F_n$ grow at most polynomially under iteration by $phi$. We restrict attention to the subset $UPG(F_n)$ of $Out(F_n)$ consisting of polynomially growing elements whose action on $H_1(F_n, Z)$ is unipotent. In particular, if $phi$ is polynomially growing and acts trivially on $H_1(F_n,Z_3)$ then $phi$ is in $UPG(F_n)$ and also every polynomially growing element of $Out(F_n)$ has a positive power that is in $UPG(F_n)$. In this paper we solve the conjugacy problem for $UPG(F_n)$. Specifically we construct an algorithm that takes as input $phi, psiin UPG(F_n)$ and outputs YES or NO depending on whether or not there is $thetain Out(F_n)$ such that $psi=thetaphitheta^{-1}$. Further, if YES then such a $theta$ is produced.
We give a short proof of Masbaum and Reids result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.
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