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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 (mathematics)|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 ''f<sup>n</sup>''(''e'') is ''immersed'', that is ''f<sup>n</sup>''(''e'') is locally injective on ''e''. Train-track maps are a key tool in analyzing the dynamics of [[automorphism]]s of [[finitely generated group|finitely generated]] [[free group]]s and in the study of the [[Marc Culler|Culler]]–[[Karen Vogtmann|Vogtmann]] [[Outer space (mathematics)|Outer space]].
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| Train track maps for free group automorphisms were introduced in a 1992 paper of [[Mladen Bestvina|Bestvina]] and Handel.<ref name="BH92">Mladen Bestvina, and Michael Handel, [http://www.jstor.org/pss/2946562 ''Train tracks and automorphisms of free groups.''] [[Annals of Mathematics]] (2), vol. 135 (1992), no. 1, pp. 1–51</ref> The notion was motivated by Thurston's [[train track (mathematics)|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 ''F<sub>n</sub>'' 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<ref name="BH92"/> the ''Scott conjecture'' which says that for every automorphism ''α'' of a finitely generated [[free group]] ''F<sub>n</sub>'' the fixed subgroup of ''α'' is free of [[rank of a group|rank]] at most ''n''. In a subsequent paper<ref name="BH95">Mladen Bestvina and Michael Handel. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-3YMWGDB-6&_user=10&_coverDate=01%2F31%2F1995&_rdoc=7&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235676%231995%23999659998%23164299%23FLP%23display%23Volume)&_cdi=5676&_sort=d&_docanchor=&_ct=14&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=95a1de9855052e28e90d58a5cd2f93b6 ''Train-tracks for surface homeomorphisms.'']
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| [[Topology (journal)|Topology]], vol. 34 (1995), no. 1, pp. 109–140.</ref> Bestvina and Handel applied the train track techniques to obtain an effective proof of Thurston's classification of [[homeomorphism]]s of compact surfaces (with or without boundary) which says that every such [[homeomorphism]] is, up to [[Homotopy#Isotopy|isotopy]], either reducible, of finite order or [[Pseudo-Anosov map|pseudo-anosov]].
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| 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(''F<sub>n</sub>''). 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 ''F<sub>n</sub>'' applied to a particular [[conjugacy class]] in ''F<sub>n</sub>''. This information is especially helpful when studying the dynamics of the action of elements of Out(''F<sub>n</sub>'') on the Culler–Vogtmann Outer space and its boundary and when studying ''F<sub>n</sub>'' actions of on [[real tree]]s.<ref>M. Bestvina, M. Feighn, M. Handel, [http://www.springerlink.com/content/50hq64n0l6gpuukk/ ''Laminations, trees, and irreducible automorphisms of free groups.''] [[Geometric and Functional Analysis]], vol. 7 (1997), no. 2, 215–244</ref><ref name="LL03">Gilbert Levitt and Martin Lustig, ''Irreducible automorphisms of ''F''<sub>''n''</sub> have north-south dynamics on compactified outer space.'' Journal of the Institute of Mathematics of Jussieu, vol. 2 (2003), no. 1, 59–72</ref><ref>Gilbert Levitt, and Martin Lustig, [http://www.reference-global.com/doi/abs/10.1515/CRELLE.2008.038 ''Automorphisms of free groups have asymptotically periodic dynamics.''] [[Crelle's journal]], vol. 619 (2008), pp. 1–36</ref> Examples of applications of train tracks include: a theorem of Brinkmann<ref name="Br">P. Brinkmann, [http://www.springerlink.com/content/xh7fpjrgerceurtw/ ''Hyperbolic automorphisms of free groups.''] [[Geometric and Functional Analysis]], vol. 10 (2000), no. 5, pp. 1071–1089</ref> proving that for an automorphism ''α'' of ''F<sub>n</sub>'' the mapping torus group of ''α'' is [[word-hyperbolic group|word-hyperbolic]] if and only if ''α'' has no periodic conjugacy classes; a theorem of Bridson and Groves<ref name="BG">Martin R. Bridson and Daniel Groves. [http://people.maths.ox.ac.uk/~bridson/papers/BG1/ ''The quadratic isoperimetric inequality for mapping tori of free-group automorphisms.''] Memoirs of the American Mathematical Society, to appear.</ref> that for every automorphism ''α'' of ''F<sub>n</sub>'' the mapping torus group of ''α'' satisfies a quadratic [[Dehn function|isoperimetric inequality]]; a proof of algorithmic solvability of the [[conjugacy problem]] for free-by-cyclic groups;<ref name="BMMV">O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=471663 ''The conjugacy problem is solvable in free-by-cyclic groups.''] [[Bulletin of the London Mathematical Society]], vol. 38 (2006), no. 5, pp. 787–794</ref> and others.
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| Train tracks were a key tool in the proof by Bestvina, Feighn and Handel that the group Out(''F<sub>n</sub>'') satisfies the [[Tits alternative]].<ref name="BFH00">Mladen Bestvina, Mark Feighn, and Michael Handel. [http://www.emis.ams.org/journals/Annals/151_2/bestvina.pdf ''The Tits alternative for Out(''F''<sub>''n''</sub>). I. Dynamics of exponentially-growing automorphisms.''] [[Annals of Mathematics]] (2), vol. 151 (2000), no. 2, pp. 517–623</ref><ref name="BFH05">Mladen Bestvina, Mark Feighn, and Michael Handel. [http://annals.princeton.edu/annals/2005/161-1/p01.xhtml ''The Tits alternative for Out(F<sub>n</sub>). II. A Kolchin type theorem.''] [[Annals of Mathematics]] (2), vol. 161 (2005), no. 1, pp. 1–59</ref>
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| The machinery of train tracks for injective [[endomorphism]]s of [[free group]]s was later developed by Dicks and Ventura.<ref name="DV">Warren Dicks, and Enric Ventura. [http://books.google.com/books?id=3sWSRRfNFKgC&pg=PP1&dq=Warren+Dicks,+and+Enric+Ventura.+%22The+group+fixed+by+a+family+of+injective+endomorphisms+of+a+free+group.%22+Contemporary+Mathematics ''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</ref>
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| ==Formal definition==
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| ===Combinatorial map===
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| For a finite graph ''Γ'' (which is thought of here as a 1-dimensional [[cell complex]]) a ''combinatorial map'' is a continuous map
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| :''f'' : ''Γ'' → ''Γ''
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| such that:
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| * The map ''f'' takes vertices to vertices.
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| * For every edge ''e'' of ''Γ'' its image ''f''(''e'') is a nontrivial edge-path ''e''<sub>1</sub>...''e''<sub>''m''</sub> in ''Γ'' where ''m'' ≥ 1. Moreover, ''e'' can be subdivided into ''m'' intervals such that the interior of the ''i''-th interval is mapped by ''f'' homeomorphically onto the interior of the edge ''e''<sub>''i''</sub> for ''i'' = 1,...,''m''.
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| ===Train track map===
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| Let ''Γ'' be a finite connected graph. A combinatorial map ''f'' : ''Γ'' → ''Γ'' is called a ''train track map'' if for every edge ''e'' of ''Γ'' and every integer ''n'' ≥ 1 the edge-path ''f''<sup>''n''</sup>(''e'') contains no backtracks, that is, it contains no subpaths of the form ''hh''<sup>−1</sup> where ''h'' is an edge of ''Γ''. In other words, the restriction of ''f''<sup>''n''</sup> to ''e'' is locally injective (or an immersion) for every edge ''e'' and every ''n'' ≥ 1.
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| When applied to the case ''n'' = 1, this definition implies, in particular, that the path ''f''(''e'') has no backtracks.
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| ===Topological representative===
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| Let ''F''<sub>''k''</sub> be a [[free group]] of finite rank ''k'' ≥ 2. Fix a free basis ''A'' of ''F''<sub>''k''</sub> and an identification of ''F''<sub>''k''</sub> with the [[fundamental group]] of the ''rose'' ''R''<sub>''k''</sub> which is a wedge of ''k'' circles corresponding to the basis elements of ''A''.
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| Let ''φ'' ∈ Out(''F''<sub>''k''</sub>) be an outer automorphism of ''F''<sub>''k''</sub>.
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| A ''topological representative'' of ''φ'' is a triple (''τ'', ''Γ'', ''f'') where:
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| *''Γ'' is a finite connected graph with the first [[betti number]] ''k'' (so that the [[fundamental group]] of ''Γ'' is free of rank ''k'').
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| * ''τ'' : ''R<sub>k</sub>'' → ''Γ'' is a [[homotopy equivalence]] (which, in this case, means that ''τ'' is a continuous map which induces an isomorphism at the level of fundamental groups).
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| * ''f'' : ''Γ'' → ''Γ'' is a combinatorial map which is also a homotopy equivalence.
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| * If ''σ'' : ''Γ'' → ''R<sub>k</sub>'' is a homotopy inverse of ''τ'' then the composition
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| : ''σfτ'' : ''R<sub>k</sub>'' → ''R<sub>k</sub>''
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| :induces an automorphism of ''F''<sub>''k''</sub> = ''π''<sub>1</sub>(''R<sub>k</sub>'') whose outer automorphism class is equal to ''φ''.
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| The map ''τ'' in the above definition is called a ''marking'' and is typically suppressed when topological representatives are discussed. Thus, by abuse of notation, one often says that in the above situation ''f'' : ''Γ'' → ''Γ'' is a topological representative of ''φ''.
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| ===Train track representative===
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| Let ''φ'' ∈ Out(''F''<sub>''k''</sub>) be an outer automorphism of ''F''<sub>''k''</sub>. A train track map which is a topological representative of ''φ'' is called a ''train track representative'' of ''φ''.
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| ===Legal and illegal turns===
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| Let ''f'' : ''Γ'' → ''Γ'' be a combinatorial map. A ''turn'' is an unordered pair ''e'', ''h'' of oriented edges of ''Γ'' (not necessarily distinct) having a common initial vertex. A turn ''e'', ''h'' is ''degenerate'' if ''e'' = ''h'' and ''nondegenerate'' otherwise.
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| A turn ''e'', ''h'' is ''illegal'' if for some ''n'' ≥ 1 the paths ''f''<sup>''n''</sup>(''e'') and ''f''<sup>''n''</sup>(''h'') have a nontrivial common initial segment (that is, they start with the same edge). A turn is ''legal'' if it not ''illegal''.
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| An edge-path ''e''<sub>1</sub>,..., ''e''<sub>''m''</sub> is said to ''contain'' turns ''e''<sub>''i''</sub><sup>−1</sup>, ''e''<sub>''i+1''</sub> for ''i'' = 1,...,''m''−1.
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| A combinatorial map ''f'' : ''Γ'' → ''Γ'' is a train-track map if and only if for every edge ''e'' of ''Γ'' the path ''f''(''e'') contains no illegal turns.
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| ===Derivative map===
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| Let ''f'' : ''Γ'' → ''Γ'' be a combinatorial map and let ''E'' be the set of oriented edges of ''Γ''. Then ''f'' determines its ''derivative map'' ''Df'' : ''E'' → ''E'' where for every edge ''e'' ''Df''(''e'') is the initial edge of the path ''f''(''e''). The map ''Df'' naturally extends to the map ''Df'' : ''T'' → ''T'' where ''T'' is the set of all turns in ''Γ''. For a turn ''t'' given by an edge-pair ''e'', ''h'', its image ''Df''(''t'') is the turn ''Df''(''e''), ''Df''(''h''). A turn ''t'' is legal if and only if for every ''n'' ≥ 1 the turn (''Df'')<sup>''n''</sup>(''t'') is nondegenerate. Since the set ''T'' of turns is finite, this fact allows one to algorithmically determine if a given turn is legal or not and hence to algorithmically decide, given ''f'', whether or not ''f'' is a train-track map.
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| ==Examples==
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| Let ''φ'' be the automorphism of ''F''(''a'',''b'') given by ''φ''(''a'') = ''b'', ''φ''(''b'') = ''ab''. Let ''Γ'' be the wedge of two loop-edges ''E''<sub>''a''</sub> and ''E''<sub>''b''</sub> corresponding to the free basis elements ''a'' and ''b'', wedged at the vertex ''v''. Let ''f'' : ''Γ'' → ''Γ'' be the map which fixes ''v'' and sends the edge ''E''<sub>''a''</sub> to ''E''<sub>''b''</sub> and that sends the edge ''E''<sub>''b''</sub> to the edge-path ''E''<sub>''a''</sub>''E''<sub>''b''</sub>.
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| Then ''f'' is a train track representative of ''φ''.
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| ==Main result for irreducible automorphisms==
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| ===Irreducible automorphisms===
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| An outer automorphism ''φ'' of ''F''<sub>''k''</sub> is said to be ''reducible'' if there exists a free product decomposition
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| :<math>F_k=H_1\ast\dots H_m\ast U</math>
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| where all ''H''<sub>''i''</sub> are nontrivial, where ''m'' ≥ 1 and where ''φ'' permutes the conjugacy classes of ''H''<sub>1</sub>,...,''H''<sub>''m''</sub> in ''F''<sub>''k''</sub>. An outer automorphism ''φ'' of ''F''<sub>''k''</sub> is said to be ''irreducible'' if it is not reducible.
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| It is known<ref name="BH92"/> that ''φ'' ∈ Out(''F''<sub>''k''</sub>) be irreducible if and only if for every topological representative
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| ''f'' : ''Γ'' → ''Γ'' of ''φ'', where ''Γ'' is finite, connected and without degree-one vertices, any proper ''f''-invariant subgraph of ''Γ'' is a forest.
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| ===Bestvina–Handel theorem for irreducible automorphisms===
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| The following result was obtained by Bestvina and Handel in their 1992 paper<ref name="BH92"/> where train track maps were originally introduced:
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| Let ''φ'' ∈ Out(''F''<sub>''k''</sub>) be irreducible. Then there exists a train track representative of ''φ''.
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| ====Sketch of the proof====
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| For a topological representative ''f'':''Γ''→''Γ'' of an automorphism ''φ'' of ''F''<sub>''k''</sub> the ''transition matrix'' ''M''(''f'') is an ''r''x''r'' matrix (where ''r'' is the number of topological edges of ''Γ'') where the entry ''m''<sub>''ij''</sub> is the number of times the path ''f''(''e''<sub>''j''</sub>) passes through the edge ''e''<sub>''i''</sub> (in either direction). If ''φ'' is irreducible, the transition matrix ''M''(''f'') is [[Irreducible (mathematics)|''irreducible'']] in the sense of the [[Perron–Frobenius theorem]] and it has a unique [[Perron–Frobenius theorem|Perron–Frobenius eigenvalue]] ''λ''(''f'') ≥ 1 which is equal to the spectral radius of ''M''(''f'').
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| One then defines a number of different ''moves'' on topological representatives of ''φ'' that are all seen to either decrease or preserve the [[Perron–Frobenius theorem|Perron–Frobenius eignevalue]] of the transition matrix. These moves include: subdividing an edge; valence-one homotopy (getting rid of a degree-one vertex); valence-two homotopy (getting rid of a degree-two vertex); collapsing an invariant forest; and folding. Of these moves the valence-one homotopy always reduced the Perron–Frobenius eigenvalue.
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| Starting with some topological representative ''f'' of an irreducible automorphism ''φ'' one then algorithmically constructs a sequence of topological representatives
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| :''f'' = ''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>,...
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| of ''φ'' where ''f''<sub>''n''</sub> is obtained from ''f''<sub>''n''−1</sub> by several moves, specifically chosen. In this sequence, if ''f''<sub>''n''</sub> is not a train track map, then the moves producing ''f''<sub>''n''+1</sub> from ''f''<sub>''n''</sub> necessarily involve a sequence of folds followed by a valence-one homotopy, so that the Perron–Frobenius eignevalue of ''f''<sub>''n''+1</sub> is strictly smaller than that of ''f''<sub>''n''</sub>. The process is arranged in such a way that Perron–Frobenius eignevalues of the maps ''f''<sub>''n''</sub> take values in a discrete substet of <math>\mathbb R</math>. This guarantees that the process terminates in a finite number of steps and the last term ''f''<sub>''N''</sub> of the sequence is a train track representative of ''φ''.
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| ====Applications to growth====
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| A consequence (requiring additional arguments) of the above theorem is the following:<ref name="BH92"/>
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| *If ''φ'' ∈ Out(''F''<sub>''k''</sub>) is irreducible then the Perron–Frobenius eigenvalue ''λ''(''f'') does not depend on the choice of a train track representative ''f'' of ''φ'' but is uniquely determined by ''φ'' itself and is denoted by ''λ''(''φ''). The number ''λ''(''φ'') is called the ''growth rate'' of ''φ''.
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| *If ''φ'' ∈ Out(''F''<sub>''k''</sub>) is irreducible and of infinite order then ''λ''(''φ'') > 1. Moreover, in this case for every free basis ''X'' of ''F''<sub>''k''</sub> and for every ''w'' ∈ ''F''<sub>''k''</sub> there exists ''C'' ≥ 1 such that for all ''n'' ≥ 1
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| :<math>\frac{1}{C}\lambda^n(\phi) \le ||\phi^n(w)||_X\le C \lambda^n(\phi), </math>
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| :where ||''u''||<sub>''X''</sub> is the cyclically reduced length of an element ''u'' of ''F''<sub>''k''</sub> with respect to ''X''.
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| Unlike for elements of [[mapping class group]]s, for an irreducible ''φ'' ∈ Out(''F''<sub>''k''</sub>) it is often the case
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| <ref name="HM07">Michael Handel, and Lee Mosher, ''The expansion factors of an outer automorphism and its inverse.''
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| [[Transactions of the American Mathematical Society]], vol. 359 (2007), no. 7, 3185 3208</ref> that
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| :''λ''(''φ'') ≠ ''λ''(''φ''<sup>−1</sup>).
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| ==Relative train tracks==
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| {{Empty section|date=July 2010}}
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| ==Applications and generalizations==
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| *The first major application of train tracks was given in the original 1992 paper of Bestvina and Handel<ref name="BH92"/> where train tracks were introduced. The paper gave a proof of the ''Scott conjecture'' which says that for every automorphism ''α'' of a finitely generated [[free group]] ''F<sub>n</sub>'' the fixed subgroup of ''α'' is free of rank at most ''n''.
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| *In a subsequent paper<ref name="BH95"/> Bestvina and Handel applied the train track techniques to obtain an effective proof of Thurston's classification of [[homeomorphism]]s of compact surfaces (with or without boundary) which says that every such [[homeomorphism]] is, up to [[Homotopy#Isotopy|isotopy]], is either reducible, of finite order or [[Pseudo-Anosov map|pseudo-anosov]].
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| *Train tracks are the main tool in Los' algorithm for deciding whether or not two irreducible elements of Out(''F<sub>n</sub>'') are [[conjugacy class|conjugate]] in Out(''F<sub>n</sub>'').<ref>Jérôme E. Los, [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-43B9F5R-D&_user=10&_coverDate=07%2F31%2F1996&_rdoc=12&_fmt=high&_orig=browse&_srch=doc-info(%23toc%235676%231996%23999649996%23253962%23FLP%23display%23Volume)&_cdi=5676&_sort=d&_docanchor=&_ct=14&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=d6139b585a9f83d728010ca53e75e286 ''On the conjugacy problem for automorphisms of free groups.''] [[Topology (journal)|Topology]], vol. 35 (1996), no. 3, pp. 779–806</ref>
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| *A theorem of Brinkmann<ref name="Br"/> proving that for an automorphism ''α'' of ''F<sub>n</sub>'' the mapping torus group of ''α'' is [[word-hyperbolic group|word-hyperbolic]] if and only if ''α'' has no periodic conjugacy classes.
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| *A theorem of Levitt and Lustig showing that a fully irreducible automorphism of a ''F''<sub>''n''</sub> has "north-south" dynamics when acting on the Thurston-type compactification of the [[Culler–Vogtmann Outer space]].<ref name="LL03"/>
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| *A theorem of Bridson and Groves<ref name="BG"/> that for every automorphism ''α'' of ''F<sub>n</sub>'' the mapping torus group of ''α'' satisfies a quadratic [[Dehn function|isoperimetric inequality]].
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| *The proof by Bestvina, Feighn and Handel that the group Out(''F<sub>n</sub>'') satisfies the [[Tits alternative]].<ref name="BFH00"/><ref name="BFH05"/>
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| *An algorithm that, given an automorphism ''α'' of ''F''<sub>''n''</sub>, decides whether or not the fixed subgroup of ''α'' is trivial and finds a finite generating set for that fixed subgroup.<ref>O. S. Maslakova. ''The fixed point group of a free group automorphism''. (Russian). Algebra Logika, vol. 42 (2003), no. 4, pp. 422–472; translation in
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| Algebra and Logic, vol. 42 (2003), no. 4, pp. 237–265</ref>
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| *The proof of algorithmic solvability of the [[conjugacy problem]] for free-by-cyclic groups by Bogopolski, Martino, Maslakova, and Ventura.<ref name="BMMV"/>
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| *The machinery of train tracks for injective [[endomorphism]]s of [[free group]]s, generalizing the case of automorphisms, was developed in a 1996 book of Dicks and Ventura.<ref name="DV"/>
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| ==See also==
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| *[[Geometric group theory]]
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| *[[Real tree]]
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| *[[Mapping class group]]
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| *[[Free group]]
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| *[[Out(Fn)|Out(''F''<sub>''n''</sub>)]]
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| ==Basic references==
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| *Mladen Bestvina, and Michael Handel, [http://www.jstor.org/pss/2946562 ''Train tracks and automorphisms of free groups.''] [[Annals of Mathematics]] (2), vol. 135 (1992), no. 1, pp. 1–51
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| *Warren Dicks, and Enric Ventura. [http://books.google.com/books?id=3sWSRRfNFKgC&pg=PP1&dq=Warren+Dicks,+and+Enric+Ventura.+%22The+group+fixed+by+a+family+of+injective+endomorphisms+of+a+free+group.%22+Contemporary+Mathematics ''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
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| *Oleg Bogopolski. [http://books.google.com/books?id=jEw8MpP6DIgC&pg=PT1&dq=Oleg+Bogopolski+Introduction+to+Group+Theory&ei=GBlzS7qVL5TyMZ7T7YsE&cd=2#v=onepage&q=Oleg%20Bogopolski%20Introduction%20to%20Group%20Theory&f=false ''Introduction to group theory''.] EMS Textbooks in Mathematics. [[European Mathematical Society]], Zürich, 2008. ISBN 978-3-03719-041-8
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| ==Footnotes==
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| {{reflist}}
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| ==External links==
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| *Peter Brinkmann's minicourse notes on train tracks [http://www.math.uiuc.edu/~brinkman/research/tex/talk1.pdf][http://www.math.uiuc.edu/~brinkman/research/tex/talk2.pdf][http://www.math.uiuc.edu/~brinkman/research/tex/talk3.pdf][http://www.math.uiuc.edu/~brinkman/research/tex/talk5.pdf]
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| [[Category:Geometric group theory]]
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| [[Category:Geometric topology]]
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| [[Category:Combinatorics on words]]
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