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arxiv: 1707.07095 · v1 · pith:B4WNEQSQnew · submitted 2017-07-22 · 🧮 math.GR · math.DS· math.GT

Counting Conjugacy Classes in Out(F_N)

classification 🧮 math.GR math.DSmath.GT
keywords classesconjugacyballcayleyexponentiallygraphgrowsnumber
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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 $N\ge 3$ 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 $\log\lambda(\phi)$ on the order of $R$ grows exponentially in $R$.

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