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The quadratic isoperimetric inequality for mapping tori of free group automorphisms
About this Title
Martin R. Bridson, Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, United Kingdom and Daniel Groves, University of Illinois at Chicago, 2851 S. Morgan St., Chicago, Illinois 60607-7045
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 203, Number 955
ISBNs: 978-0-8218-4631-5 (print); 978-1-4704-0569-4 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00578-X
Published electronically: August 27, 2009
Keywords: Free-by-cyclic groups,
automorphisms of free groups,
isoperimetric inequalities,
Dehn functions
MSC: Primary 20F65, (20F06, 20E36, 57M07)
Table of Contents
Chapters
- Introduction
1. Positive Automorphisms
2. Train Tracks and the Beaded Decomposition
3. The General Case
Abstract
We prove that if $F$ is a finitely generated free group and $\phi$ is an automorphism of $F$ then $F\rtimes _\phi \mathbb Z$ satisfies a quadratic isoperimetric inequality.
Our proof of this theorem rests on a direct study of the geometry of van Kampen diagrams over the natural presentations of free-by-cylic groups. The main focus of this study is on the dynamics of the time flow of $t$-corridors, where $t$ is the generator of the $\mathbb Z$ factor in $F\rtimes _\phi \mathbb Z$ and a $t$-corridor is a chain of 2-cells extending across a van Kampen diagram with adjacent 2-cells abutting along an edge labelled $t$. We prove that the length of $t$-corridors in any least-area diagram is bounded by a constant times the perimeter of the diagram, where the constant depends only on $\phi$. Our proof that such a constant exists involves a detailed analysis of the ways in which the length of a word $w\in F$ can grow and shrink as one replaces $w$ by a sequence of words $w_m$, where $w_m$ is obtained from $\phi (w_{m-1})$ by various cancellation processes. In order to make this analysis feasible, we develop a refinement of the improved relative train track technology due to Bestvina, Feighn and Handel.
- Juan M. Alonso, Inégalités isopérimétriques et quasi-isométries, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 12, 761–764 (French, with English summary). MR 1082628
- Mladen Bestvina, The topology of $\textrm {Out}(F_n)$, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 373–384. MR 1957048
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226
- Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for $\textrm {Out}(F_n)$. I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517–623. MR 1765705, DOI 10.2307/121043
- Mladen Bestvina, Mark Feighn, and Michael Handel, Solvable subgroups of $\textrm {Out}(F_n)$ are virtually Abelian, Geom. Dedicata 104 (2004), 71–96. MR 2043955, DOI 10.1023/B:GEOM.0000022864.30278.34
- Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for $\textrm {Out}(F_n)$. II. A Kolchin type theorem, Ann. of Math. (2) 161 (2005), no. 1, 1–59. MR 2150382, DOI 10.4007/annals.2005.161.1
- Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1–51. MR 1147956, DOI 10.2307/2946562
- O. Bogopolski, A. Martino, O. Maslakova, and E. Ventura, The conjugacy problem is solvable in free-by-cyclic groups, Bull. London Math. Soc. 38 (2006), no. 5, 787–794. MR 2268363, DOI 10.1112/S0024609306018674
- N. Brady and M.R. Bridson, On the absence of biautomaticity in certain graphs of abelian groups, preprint.
- Martin R. Bridson, Polynomial Dehn functions and the length of asynchronously automatic structures, Proc. London Math. Soc. (3) 85 (2002), no. 2, 441–466. MR 1912057, DOI 10.1112/S0024611502013564
- Martin R. Bridson, On the subgroups of semihyperbolic groups, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, 2001, pp. 85–111. MR 1929323
- Martin R. Bridson and Simon M. Salamon (eds.), Invitations to geometry and topology, Oxford Graduate Texts in Mathematics, vol. 7, Oxford University Press, Oxford, 2002. Dedicated to Brian Steer to mark his 60th birthday. MR 1967744
- M. R. Bridson and S. M. Gersten, The optimal isoperimetric inequality for torus bundles over the circle, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 1–23. MR 1380947, DOI 10.1093/qmath/47.1.1
- M.R. Bridson and D. Groves, The growth of conjugacy classes under free-group automorphisms, in preparation.
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- M.R. Bridson and L. Reeves, On the absence of automaticity in certain free-by-cyclic groups, in preparation.
- Martin R. Bridson and Karen Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 301–316. MR 2264548, DOI 10.1090/pspum/074/2264548
- P. Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 (2000), no. 5, 1071–1089. MR 1800064, DOI 10.1007/PL00001647
- P. Brinkmann, Dynamics of free group automorphisms, preprint.
- Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
- Daryl Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987), no. 2, 453–456. MR 916179, DOI 10.1016/0021-8693(87)90229-8
- David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694
- Mark Feighn and Michael Handel, Mapping tori of free group automorphisms are coherent, Ann. of Math. (2) 149 (1999), no. 3, 1061–1077. MR 1709311, DOI 10.2307/121081
- M. Feighn and M. Handel, The Recognition Theorem for Out$(F_n)$, preprint.
- S. M. Gersten, The automorphism group of a free group is not a $\textrm {CAT}(0)$ group, Proc. Amer. Math. Soc. 121 (1994), no. 4, 999–1002. MR 1195719, DOI 10.1090/S0002-9939-1994-1195719-9
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- Egbert R. Van Kampen, On Some Lemmas in the Theory of Groups, Amer. J. Math. 55 (1933), no. 1-4, 268–273. MR 1506963, DOI 10.2307/2371129
- Bernhard Leeb, $3$-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995), no. 2, 277–289. MR 1358977, DOI 10.1007/BF01231445
- M. Lustig, Structure and conjugacy for automorphisms of free groups I,II, MPI-Preprint series (2000) 241 and (2001) 4.
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- N. Macura, Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups, Geom. Funct. Anal. 10 (2000), no. 4, 874–901. MR 1791144, DOI 10.1007/PL00001642
- A. Yu. Ol′shanskii and M. V. Sapir, Groups with small Dehn functions and bipartite chord diagrams, Geom. Funct. Anal. 16 (2006), no. 6, 1324–1376. MR 2276542, DOI 10.1007/s00039-006-0580-9
- P. Papasoglu, On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality, J. Differential Geom. 44 (1996), no. 4, 789–806. MR 1438192
- Saul Schleimer, Polynomial-time word problems, Comment. Math. Helv. 83 (2008), no. 4, 741–765. MR 2442962, DOI 10.4171/CMH/142
- Z. Sela, The Nielsen-Thurston classification and automorphisms of a free group. I, Duke Math. J. 84 (1996), no. 2, 379–397. MR 1404334, DOI 10.1215/S0012-7094-96-08413-6
- E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. MR 2209438