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**Algebraic linearity for an automorphism of a surface group.**
*(English)*
Zbl 0646.57007

Let W be a finite set of (cyclic) words in a free group. When studying the question whether W can be extended to a basis of the free group, Whitehead introduced a labelled graph G(W) now known as the coinitial graph associated to W.

Now let M be a compact surface with boundary (so it has free fundamental group) and S(M) be the set of isotopy classes of families of pairwise disjoint simple loops on M. Then each \(\Lambda\in S(M)\) is represented by a set of cyclic words \(W_{\Lambda}\) in the fundamental group.

“Our first result is to show that the set of coinitial graphs of the \(W_{\Lambda}\), as \(\Lambda\) ranges over S(M), may be reinterpreted as a finite set of canonical ‘train tracks’ (in the sense of Thurston) where the labels of the graph correspond to weights on these tracks. The second and main result of this paper is a linearity theorem which asserts that if an automorphism of the fundamental group of the surface is induced by a diffeomorphism and maps a set of weights supported to one track to a set of weights supported on the same or any other track, then, with some appropriate algebraic restrictions, the action is linear on the positive linear span of the weights (this is related to Thurston’s theorem about the piecewise integral linearity of the action of pseudo-Anosov maps on S(M)).” Then similar results are discussed for the case of closed surfaces, with some stronger restrictions on the automorphism.

Now let M be a compact surface with boundary (so it has free fundamental group) and S(M) be the set of isotopy classes of families of pairwise disjoint simple loops on M. Then each \(\Lambda\in S(M)\) is represented by a set of cyclic words \(W_{\Lambda}\) in the fundamental group.

“Our first result is to show that the set of coinitial graphs of the \(W_{\Lambda}\), as \(\Lambda\) ranges over S(M), may be reinterpreted as a finite set of canonical ‘train tracks’ (in the sense of Thurston) where the labels of the graph correspond to weights on these tracks. The second and main result of this paper is a linearity theorem which asserts that if an automorphism of the fundamental group of the surface is induced by a diffeomorphism and maps a set of weights supported to one track to a set of weights supported on the same or any other track, then, with some appropriate algebraic restrictions, the action is linear on the positive linear span of the weights (this is related to Thurston’s theorem about the piecewise integral linearity of the action of pseudo-Anosov maps on S(M)).” Then similar results are discussed for the case of closed surfaces, with some stronger restrictions on the automorphism.

Reviewer: B.Zimmermann

### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

57M05 | Fundamental group, presentations, free differential calculus |

### Keywords:

action on the positive linear span of weights; automorphism of the fundamental group of a surface; compact surface with boundary; isotopy classes of families of pairwise disjoint simple loops; coinitial graphs; train tracks
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\textit{J. S. Birman} and \textit{C. Series}, J. Pure Appl. Algebra 52, No. 3, 227--275 (1988; Zbl 0646.57007)

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### References:

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