By Andrew J. Casson

This e-book, which grew out of Steven Bleiler's lecture notes from a direction given by means of Andrew Casson on the collage of Texas, is designed to function an advent to the purposes of hyperbolic geometry to low dimensional topology. particularly it presents a concise exposition of the paintings of Neilsen and Thurston at the automorphisms of surfaces. The reader calls for merely an realizing of easy topology and linear algebra, whereas the early chapters on hyperbolic geometry and geometric buildings on surfaces can profitably be learn by way of a person with an information of normal Euclidean geometry meaning to research extra abour different 'geometric structures'.

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**Additional resources for Automorphisms of Surfaces after Nielsen and Thurston**

**Example text**

M is a closed V, n -manifold. theory can be made into 'coefficients Z ' by adding orientations and an i n j 3. There is a natural way of regarding W x M as a V-manifold, untwisted neighbourhood for the singularity (see Chapter ill, Example; .. ,oreach U-mamfold W (e. g. by relabelhng or forgettmg some structure). •. Then the theory VIU x M is defined by considering polyhedra P ting Z -homology (see §S). I .... f1tha two stage stratIfIcatIOn P::) S(P) and extra structure such that: j. 3. £0 = {p'l, £1 = {xix ~ SOJ, £n is all closed £n-l- manifolds.

Proof of exactness in W=_(SWXbP+lL(bP+l, 1. 2. ) p) '-------" MXI v(-S(M) Proof of Proposition 2. 5 (continued). Now let us look at the image of [SM] through the morphism _ I/> : P nn-p I8iF p id I8iI/>p nn-p I8iF p-l . i i C of Jt~e~oundar~ ~f ~he cdmPlement of a regular neighbourhood of V in the (n-p+ I)-stratum and a (p, n)-manifold n - p + 1. The singularities M' 8M have been resolved up to bardism. ] I8i})\J-I: this is nothing else than the bordism See Fig, 11. ] n n-p such that p+l ~P+l[SW] = [SM], Suppose first that SW is a set of components all labelled by bP+1 € BP+1, We can always reduce to the case M x II [SW] € (SW)kI8iiJ'+I and gets the desired manifold M'.

G' + g +, .. + g (g~ E G'; g. E G - G'). " + g, Therefore we attach a sheet q 1 (V x I) ® g' to W r' = g' +", ] along V and change the label r into a + g' + g', which is now a relation in G', p -- Let V ® r, V x (ii) component 'I' ®; r, V ® r, '" ... q,l r 2 / 1 r' ... r' l/~;h{" G-bordism Wand remove from it all the top dimensional strata which We show how to make W a G'-bordism by inserting new sheets. q,2 r N' To this purpose we reconsider the G'-manifold in general. - If [M] dl n(-; G), Then I/I(M) is a (G', n)-manifold Un(-; G').