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Out(F3) Index Realization

Published online by Cambridge University Press:  05 October 2015

CATHERINE PFAFF
CATHERINE PFAFF*
Affiliation:
Mathematics Department, UC Santa Barbara, Santa Barbara, CA 93106. e-mail: [email protected]
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Abstract

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this paper we determine an analog to the theorem for Out(F3). That is, we determine which index lists permitted by the [GJLL98] index sum inequality are achieved by ageometric fully irreducible outer automorphisms of the rank-3 free group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 445 - 458
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[BF94] Bestvina, M. and Feighn, M. Outer Limits. preprint (1994), 1–19.Google Scholar
[BFH97] Bestvina, M., Feighn, M. and Handel, M. Laminations, trees and irreducible automorphisms of free groups. Geom. Funct. Anal. 7 (1997), no. 2, 215–244.Google Scholar
[BFH00] Bestvina, M., Feighn, M. and Handel, M. The Tits alternative for Out (Fn ) I: dynamics of exponentially-growing automorphisms. Ann. of Math.-Se. 151 (2000), no. 2, 517–624.Google Scholar
[BH92] Bestvina, M. and Handel, M. Train tracks and automorphisms of free groups. Ann. of Math. 135 (1992), no. 1, 1–51.Google Scholar
[CH] Coulbois, T. and Hilion, A. Rips induction: index of the dual lamination of an R-tree. (2010), arXiv preprint arXiv:1002.0972.Google Scholar
[CH12] Coulbois, T. and Hilion, A. Botany of irreducible automorphisms of free groups. Pacific J. Math. 256 (2012), no. 2.Google Scholar
[Coo87] Cooper, D. Automorphisms of free groups have finitely generated fixed point sets. J. Algebra 111 (1987), no. 2, 453–456.Google Scholar
[Cou14] Coulbois, T. Free group automorphisms and train-track representative in python/sage. https://github.com/coulbois/sage-train-track (2012–2014).Google Scholar
[CV86] Culler, M. and Vogtmann, K. Moduli of graphs and automorphisms of free groups. Invent. math. 84 (1986), no. 1, 91–119.Google Scholar
[DPS75] Dyer, J. L and Scott, G. P. Periodic automorphisms of free groups. Comm. Algebra 3 (1975), no. 3, 195–201.Google Scholar
[EMR12] Eskin, A., Mirzakhani, M. and Rafi, K. Counting closed geodesics in strata. arXiv preprint arXiv:1206.5574 (2012).Google Scholar
[FdPDdm91] Fathi, A. and Université de Paris 11. Département de mathématique. Travaux de Thurston sur les surfaces: Séminaire Orsay [1991–1979]. (Société mathématique de France, 1991).Google Scholar
[FH11] Feighn, M. and Handel, M. The Recognition Theorem for Out (Fn ). Groups Geom. Dyn. 5 (2011), no. 1, 39–106. MR 2763779 (2012b:20061)Google Scholar
[Ger87] Gersten, S.M. Fixed points of automorphisms of free groups. Math. 64 (1987), no. 1, 51–85.Google Scholar
[GJLL98] Gaboriau, D., Jäger, A., Levitt, G. and Lustig, M. An index for counting fixed points of automorphisms of free groups. Duke math. J. 93 (1998), no. 3, 425–452.Google Scholar
[GL95] Gaboriau, D. and Levitt, G. The rank of actions on R-trees. Ann. Sci. l'Ecole norm. sup. vol. 28 (Société mathématique de France, 1995), pp. 549–570.Google Scholar
[Gui05] Guirardel, Vincent Cœur et nombre d'intersection pour les actions de groupes sur les arbres. Ann. Sci. l'École Norm. Sup. vol. 38 (Elsevier, 2005), pp. 847–888.Google Scholar
[HM07] Handel, M. and Mosher, L. Parageometric outer automorphisms of free groups. Trans Amer. Math. Soc. 359 (2007), no. 7, 3153–3184.Google Scholar
[HM11] Handel, M. and Mosher, L. Axes in Outer Space. Amer Math. Soc. (2011), no. 1004.Google Scholar
[JL09] Jäger, A. and Lustig, M. Free group automorphisms with many fixed points at infinity, arXiv preprint arXiv:0904.1533 (2009).Google Scholar
[KZ03] Kontsevich, M. and Zorich, A. Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. math. 153 (2003), no. 3, 631–678.Google Scholar
[Lan04] Lanneau, E. Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities. Comment. Math. Helv. 79 (2004), no. 3, 471–501.Google Scholar
[Lan05] Lanneau, E. Connected components of the strata of the moduli spaces of quadratic differentials. arXiv preprint math/0506136 (2005).Google Scholar
[LL03] Levitt, G. and Lustig, M. Irreducible automorphisms of Fn have north–south dynamics on compactified outer space. J. Inst. Math. Jussieu. 2 (2003), no. 01, 59–72.Google Scholar
[Mah11] Maher, J. Random walks on the mapping class group. Duke Math. J. 156 (2011), no. 3, 429–468.Google Scholar
[MS93] Masur, H. and Smillie, J. Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms. Comm. Math. Helv. 68 (1993), no. 1, 289–307.Google Scholar
[Pfa12a] Pfaff, C. Constructing and Classifying Fully Irreducible Outer Automorphisms of Free Groups. Ph.D. thesis, Rutgers University, 2012.Google Scholar
[Pfa12b] Pfaff, C. Ideal Whitehead graphs in Out(Fr ) I: some unachieved graphs. To appear in New York J. Math. arXiv preprint arXiv:1210.5762 (2012).Google Scholar
[Pfa13a] Pfaff, C. Ideal Whitehead Graphs in Out(Fr ) II: The complete graph in each rank. J. Homotopy and Related Structures 10.1007/s40062-013-0060-5 (2013).Google Scholar
[Pfa13b] Pfaff, C. Ideal Whitehead graphs in Out(Fr ) III: achieved graphs in rank 3. To appear in J. of Topology and Analysis, arXiv preprint arXiv:1301.7080 (2013).Google Scholar
[Zor10] Zorich, A. Explicit Jenkins–Strebel representatives of all strata of Abelian and quadratic differentials. arXiv preprint arXiv:1011.0395 (2010).Google Scholar