Combinatorial mapping-torus, branched surfaces and free group automorphisms

François Gautero

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 3, page 405-440
  • ISSN: 0391-173X

Abstract

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We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a 1 -cocycle of a 2 -complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates the link between this new representation (first introduced in [16]) and the classical representation of a free group automorphism by a graph-map [2].

How to cite

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Gautero, François. "Combinatorial mapping-torus, branched surfaces and free group automorphisms." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 6.3 (2007): 405-440. <http://eudml.org/doc/272278>.

@article{Gautero2007,
abstract = {We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a $1$-cocycle of a $2$-complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates the link between this new representation (first introduced in [16]) and the classical representation of a free group automorphism by a graph-map [2].},
author = {Gautero, François},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {geometric automorphisms; free group automorphisms; pseudo-Anosov homeomorphisms; outer automorphisms; dynamical branched surfaces; fundamental groups; mapping-torus groups},
language = {eng},
number = {3},
pages = {405-440},
publisher = {Scuola Normale Superiore, Pisa},
title = {Combinatorial mapping-torus, branched surfaces and free group automorphisms},
url = {http://eudml.org/doc/272278},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Gautero, François
TI - Combinatorial mapping-torus, branched surfaces and free group automorphisms
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 3
SP - 405
EP - 440
AB - We give a characterization of the geometric automorphisms in a certain class of (not necessarily irreducible) free group automorphisms. When the automorphism is geometric, then it is induced by a pseudo-Anosov homeomorphism without interior singularities. An outer free group automorphism is given by a $1$-cocycle of a $2$-complex (a standard dynamical branched surface, see [7] and [9]) the fundamental group of which is the mapping-torus group of the automorphism. A combinatorial construction elucidates the link between this new representation (first introduced in [16]) and the classical representation of a free group automorphism by a graph-map [2].
LA - eng
KW - geometric automorphisms; free group automorphisms; pseudo-Anosov homeomorphisms; outer automorphisms; dynamical branched surfaces; fundamental groups; mapping-torus groups
UR - http://eudml.org/doc/272278
ER -

References

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  1. [1] M. Bestvina, M. Feighn and M. Handel, The Tits alternative for Out ( F n ) : Dynamics of exponentially growing free group automorphisms, Ann. of Math.151 (2000), 517–623. Zbl0984.20025MR1765705
  2. [2] M. Bestvina and M. Handel, Train-tracks and free group-automorphisms, Ann. of Math.135 (1992), 1–52. Zbl0757.57004MR1147956
  3. [3] M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology34 (1995), 109–140. Zbl0837.57010MR1308491
  4. [4] R. Benedetti and C. Petronio, A finite graphic calculus for 3 -manifolds, Manuscripta Math.88 (1995), 291–310. Zbl0856.57009MR1359699
  5. [5] R. Benedetti and C. Petronio, “Branched Standard Spines of 3 -Manifolds", Lectures Notes in Mathematics, 1653, Springer, Berlin, 1997. Zbl0873.57002MR1470454
  6. [6] N. Brady and J. Crisp, CAT ( 0 ) and CAT ( - 1 ) dimensions of torsion free hyperbolic groups, to appear in Comment. Math. Helv. Zbl1145.20023MR2296058
  7. [7] B. G. Casler, An imbedding theorem for connected 3 -manifolds with boundary, Proc. Amer. Math. Soc.16 (1965), 559–566. Zbl0129.15801MR178473
  8. [8] A. J. Casson and S. A. Bleiler“Automorphisms of Surfaces After Nielsen and Thurston", London Mathematical Society Student Texts 9, Cambridge University Press, Cambridge, 1988. Zbl0649.57008MR964685
  9. [9] J. Christy, Branched surfaces and attractors I: Dynamic branched surfaces, Trans. Amer. Math. Soc.336 (1993), 759–784. Zbl0777.58024MR1148043
  10. [10] H. D. Coldewey, E. Vogt and H. Zieschang, “Surfaces and Planar Discontinuous Groups", Lecture Notes in Mathematics 835, Springer-Verlag, 1980. Zbl0438.57001MR606743
  11. [11] D. J. Collins and H. Zieschang, Combinatorial group theory and fundamental groups, In: “Algebra VII”, Encyclopaedia Math. Sci., Vol. 58, Springer, Berlin, 1993, 1–166. Zbl0781.20020MR1265270
  12. [12] W. Dicks and E. Ventura, Irreducible automorphisms of growth rate one, J. Pure Appl. Algebra88 (1993), 51–62. Zbl0787.20023MR1233313
  13. [13] A. Fathi, F. Laudenbach and V. Poenaru, “Travaux de Thurston sur les Surfaces", Astérique 66-67, 1979. Zbl0446.57010
  14. [14] J. Fehrenbach, “Quelques Aspects Géométriques et Dynamiques du Mapping-Class Group”, PhD dissertation, Université de Nice, Sophia Antipolis, 1998. 
  15. [15] D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J.93 (1998), 425–452. Zbl0946.20010MR1626723
  16. [16] F. Gautero, Dynamical 2 -complexes, Geom. Dedicata88 (2001), 283–319. Zbl1001.57006MR1877221
  17. [17] F. Gautero, Cross-sections to semi-flows on 2 -complexes, Ergod. Theory Dyn. Syst.23 (2003), 143–174. Zbl1140.37311MR1971200
  18. [18] S. Gersten, Geometric automorphisms of a free group of rank at least three are rare, Proc. Amer. Math. Soc.89 (1983), 27–31. Zbl0525.20022MR706503
  19. [19] V. Guirardel, Core and intersection number for group action on trees, Ann. Sci. Ecole Norm. Sup.38 (2005), 847–888. Zbl1110.20019MR2216833
  20. [20] M. Handel and L. Mosher, Parageometric outer automorphisms of free groups, Trans. Amer. Math. Soc.359 (2007), 3153–3183. Zbl1120.20042MR2299450
  21. [21] J. Los, On the conjugacy problem for automorphisms of free groups, With an addendum by the author, Topology 35 (1996), 779–808. Zbl0858.20022MR1396778
  22. [22] J. Los and M. Lustig, The set of train-track representatives of an irreducible free group automorphisms is contractible, http://www.crm.es/Publications/Preprints04.htm (2004). 
  23. [23] J. Los and Z. Nitecki, Edge-transitive graph automorphisms and periodic surface homeomorphisms, Internat. J. Bifur. Chaos Appl. Sci. Engrg.9 (1999), 1803–1813. Zbl1089.37522MR1728740
  24. [24] M. Lustig, Structure and conjugacy for automorphisms of free groups I, II, Max-Planck-Institut für Mathematik, Preprint Series 130 (2000) and 4 (2001). 
  25. [25] S. V. Matveev, Special spines of piecewise linear manifolds, Math. USSR Sb. 21 (1973). Zbl0289.57008
  26. [26] R. Penner and J. Harer, “Combinatorics of Train-Tracks", Annals of Mathematical Studies 125, Princeton University Press, 1991. Zbl0765.57001MR1144770
  27. [27] J. R. Stallings, Topologically unrealizable automorphisms of free groups, Proc. Amer. Math. Soc.84 (1982), 21–24. Zbl0477.20012MR633269
  28. [28] J. R. Stallings, Topology of finite graphs, Invent. Math.71 (1983), 551–565. Zbl0521.20013MR695906
  29. [29] R. F. Williams, Expanding attractors, Inst. Hautes Étud. Sci. Publ. Math. 43 (1974), 169–203. Zbl0279.58013MR348794

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