Skip to main content Accessibility help
×
Home
Hostname: page-component-7ccbd9845f-zxw8g Total loading time: 0.565 Render date: 2023-01-28T10:46:10.948Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

COUNTING CONJUGACY CLASSES IN $\text{Out}(F_{N})$

Published online by Cambridge University Press:  28 March 2018

MICHAEL HULL
Affiliation:
Department of Mathematics, University of Florida, Box 118105, Gainesville, FL 32611-8105, USA email mbhull@ufl.edu
ILYA KAPOVICH*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA email kapovich@math.uiuc.edu

Abstract

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 $R$.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The second author was supported by the individual NSF grants DMS-1405146 and DMS-1710868. Both authors acknowledge the support of the conference grant DMS-1719710 ‘Conference on Groups and Computation’.

References

Algom-Kfir, Y. and Bestvina, M., ‘Asymmetry of Outer space’, Geom. Dedicata 156 (2012), 8192.CrossRefGoogle Scholar
Antolin, Y., Mj, M., Sisto, A. and Taylor, S. J., ‘Intersection properties of stable subgroups and bounded cohomology’, Indiana University Math. J., to appear, arXiv:1612.07227.Google Scholar
Aougab, T., Durham, M. G. and Taylor, S. J., ‘Pulling back stability with applications to Out(F n ) and relatively hyperbolic groups’, J. London Math. Soc. 96 (2017), 565583.Google Scholar
Bestvina, M., ‘Geometry of Outer space’, in: Geometric Group Theory, IAS/Park City Mathematics Series, 21 (American Mathematical Society, Providence, RI, 2014), 173206.Google Scholar
Bestvina, M. and Feighn, M., ‘Hyperbolicity of the complex of free factors’, Adv. Math. 256 (2014), 104155.CrossRefGoogle Scholar
Bestvina, M. and Handel, M., ‘Train tracks and automorphisms of free groups’, Ann. of Math. 135 (1992), 151.Google Scholar
Coulbois, T. and Hilion, A., ‘Botany of irreducible automorphisms of free groups’, Pacific J. Math. 256(2) (2012), 291307.CrossRefGoogle Scholar
Coulbois, T. and Lustig, M., ‘Index realization for automorphisms of free groups’, Illinois J. Math. 59(4) (2015), 11111128.Google Scholar
Culler, M. and Vogtmann, K., ‘Moduli of graphs and automorphisms of free groups’, Invent. Math. 84(1) (1986), 91119.CrossRefGoogle Scholar
Dahmani, F., Guirardel, V. and Osin, D., ‘Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces’, Mem. Amer. Math. Soc. 245 (2017), no. 1156.Google Scholar
Dowdall, S. and Taylor, S. J., ‘Hyperbolic extensions of free groups’, Geom. Topol. 22(1) (2018), 517570.Google Scholar
Eskin, A. and Mirzakhani, M., ‘Counting closed geodesics in moduli space’, J. Mod. Dyn. 5(1) (2011), 71105.CrossRefGoogle Scholar
Francaviglia, S. and Martino, A., ‘Metric properties of Outer space’, Publ. Mat. 55(2) (2011), 433473.CrossRefGoogle Scholar
Handel, M. and Mosher, L., ‘The expansion factors of an outer automorphism and its inverse’, Trans. Amer. Math. Soc. 359 (2007), 31853208.CrossRefGoogle Scholar
Handel, M. and Mosher, L., ‘Axes in Outer space’, Mem. Amer. Math. Soc. 213 (2011), no. 1004.Google Scholar
Hull, M., ‘Small cancellation in acylindrically hyperbolic groups’, Groups Geom. Dynam. 10(4) (2016), 10771119.CrossRefGoogle Scholar
Hull, M. and Osin, D., ‘Conjugacy growth of finitely generated groups’, Adv. Math. 235 (2013), 361389.CrossRefGoogle Scholar
Kapovich, I., ‘Algorithmic detectability of iwip automorphisms’, Bull. Lond. Math. Soc. 46(2) (2014), 279290.CrossRefGoogle Scholar
Kapovich, I. and Bell, M., ‘Detecting fully irreducible automorphisms: Q polynomial time algorithm’, Exp. Math. ; doi:10.1080/10586458.2017.1326326.Google Scholar
Margulis, G. A., On Some Aspects of the Theory of Anosov Systems, Springer Monographs in Mathematics (Springer, Berlin, 2004).CrossRefGoogle Scholar
Osin, D., ‘Acylindrically hyperbolic groups’, Trans. Amer. Math. Soc. 368(2) (2016), 851888.CrossRefGoogle Scholar
Pfaff, C., ‘Ideal Whitehead graphs in Out(F r ). I. Some unachieved graphs’, New York J. Math. 21 (2015), 417463.Google Scholar
Yang, W., ‘Statistically convex-cocompact actions of groups with contracting elements’, Preprint, 2016, arXiv:1612.03648.Google Scholar
Yang, W., ‘Genericity of contracting elements in groups’, Preprint, 2017, arXiv:1707.06006.Google Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

COUNTING CONJUGACY CLASSES IN $\text{Out}(F_{N})$
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

COUNTING CONJUGACY CLASSES IN $\text{Out}(F_{N})$
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

COUNTING CONJUGACY CLASSES IN $\text{Out}(F_{N})$
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *