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COUNTING CONJUGACY CLASSES IN
$\text{Out}(F_{N})$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 28 March 2018, pp. 412-421
- Print publication:
- June 2018
-
- Article
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- You have access
- Export citation
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We show that if a finitely generated group
$G$ has a nonelementary 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 
$N\geq 3$ the number of distinct 
$\text{Out}(F_{N})$-conjugacy classes of fully irreducible elements 
$\unicode[STIX]{x1D719}$ from an 
$R$-ball in the Cayley graph of 
$\text{Out}(F_{N})$ with 
$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$ of the order of 
$R$ grows exponentially in 
$R$.
