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In the mathematical subject of geometric group theory a train track map is a continuous map ''f'' from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge ''e'' of the graph and for every positive integer ''n'' the path ''fn''(''e'') is ''immersed'', that is ''fn''(''e'') is locally injective on ''e''. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler–Vogtmann Outer space. ==History== Train track maps for free group automorphisms were introduced in a 1992 paper of Bestvina and Handel.〔Mladen Bestvina, and Michael Handel, (''Train tracks and automorphisms of free groups.'' ) Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1–51〕 The notion was motivated by Thurston's train tracks on surfaces, but the free group case is substantially different and more complicated. In their 1992 paper Bestvina and Handel proved that every irreducible automorphism of ''Fn'' has a train-track representative. In the same paper they introduced the notion of a ''relative train track'' and applied train track methods to solve〔 the ''Scott conjecture'' which says that for every automorphism ''α'' of a finitely generated free group ''Fn'' the fixed subgroup of ''α'' is free of rank at most ''n''. In a subsequent paper〔Mladen Bestvina and Michael Handel. (''Train-tracks for surface homeomorphisms.'' ) Topology, vol. 34 (1995), no. 1, pp. 109–140.〕 Bestvina and Handel applied the train track techniques to obtain an effective proof of Thurston's classification of homeomorphisms of compact surfaces (with or without boundary) which says that every such homeomorphism is, up to isotopy, either reducible, of finite order or pseudo-anosov. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(''Fn''). Train tracks are particularly useful since they allow to understand long-term growth (in terms of length) and cancellation behavior for large iterates of an automorphism of ''Fn'' applied to a particular conjugacy class in ''Fn''. This information is especially helpful when studying the dynamics of the action of elements of Out(''Fn'') on the Culler–Vogtmann Outer space and its boundary and when studying ''Fn'' actions of on real trees.〔M. Bestvina, M. Feighn, M. Handel, (''Laminations, trees, and irreducible automorphisms of free groups.'' ) Geometric and Functional Analysis, vol. 7 (1997), no. 2, 215–244〕〔Gilbert Levitt and Martin Lustig, ''Irreducible automorphisms of ''F''''n'' have north-south dynamics on compactified outer space.'' Journal of the Institute of Mathematics of Jussieu, vol. 2 (2003), no. 1, 59–72〕〔Gilbert Levitt, and Martin Lustig, (''Automorphisms of free groups have asymptotically periodic dynamics.'' ) Crelle's journal, vol. 619 (2008), pp. 1–36〕 Examples of applications of train tracks include: a theorem of Brinkmann〔P. Brinkmann, (''Hyperbolic automorphisms of free groups.'' ) Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071–1089〕 proving that for an automorphism ''α'' of ''Fn'' the mapping torus group of ''α'' is word-hyperbolic if and only if ''α'' has no periodic conjugacy classes; a theorem of Bridson and Groves〔Martin R. Bridson and Daniel Groves. (''The quadratic isoperimetric inequality for mapping tori of free-group automorphisms.'' ) Memoirs of the American Mathematical Society, to appear.〕 that for every automorphism ''α'' of ''Fn'' the mapping torus group of ''α'' satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups;〔O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, (''The conjugacy problem is solvable in free-by-cyclic groups.'' ) Bulletin of the London Mathematical Society, vol. 38 (2006), no. 5, pp. 787–794〕 and others. Train tracks were a key tool in the proof by Bestvina, Feighn and Handel that the group Out(''Fn'') satisfies the Tits alternative.〔Mladen Bestvina, Mark Feighn, and Michael Handel. (''The Tits alternative for Out(''F''''n''). I. Dynamics of exponentially-growing automorphisms.'' ) Annals of Mathematics (2), vol. 151 (2000), no. 2, pp. 517–623〕〔Mladen Bestvina, Mark Feighn, and Michael Handel. (''The Tits alternative for Out(Fn). II. A Kolchin type theorem.'' ) Annals of Mathematics (2), vol. 161 (2005), no. 1, pp. 1–59〕 The machinery of train tracks for injective endomorphisms of free groups was later developed by Dicks and Ventura.〔Warren Dicks, and Enric Ventura. (''The group fixed by a family of injective endomorphisms of a free group.'' ) Contemporary Mathematics, 195. American Mathematical Society, Providence, RI, 1996. ISBN 0-8218-0564-9〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Train track map」の詳細全文を読む スポンサード リンク
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