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**Additional resources for An introduction to Lorentz surfaces**

**Sample text**

By adding small open neighbourhoods in X of each point in ∂Ui , we obtain an open subset Vi of X such that Ui is closed in Vi . We now find thickenings of U1 ∪ · · · ∪ Ui by induction on i. What we have to show is that if a (G, X)-manifold M (with boundary) is the union of two open submanifolds U1 and U2 , and if U1 has a thickening U1∗ and U2 has a thickening U2∗ , then M also has a thickening. To see this, we write X1 = M\U1 and X2 = M\U2 . Then X1 and X2 are disjoint closed subsets of M. Let X1 ⊂ V1 and X2 ⊂ V2 , where V1 and V2 are open in M, V1 ∩ V2 = Ø, V1 ⊂ U1 and V2 ⊂ U2 .

Definition. A map f : M1 → M2 between Riemannian manifolds is an isometric map if it takes rectifiable paths to rectifiable paths of the same length. A path space is a path connected metric space in which the metric is determined by path length. 4. 4 In a path space, there is not necessarily a path between a and b whose length is d(a, b). For example the punctured disk D2 \{0} is a path space, in which there exist such a and b. The annulus (see the above figure), with the subspace metric induced from R2 , is not a path space.

I,” Annals of Math. 91(1970), 570–600. [18] L. Bers, “Spaces of Kleinian groups”, in Maryland Conference in Several Complex Variables I. Springer-Verlag Lecture Notes in Math, vol. 155(1970), 9–34. [19] L. Bers, “On moduli of Kleinian groups,” Russian Math Surveys 29(1974), 88– 102. [20] M. Bestvina, “Degenerations of the hyperbolic space,” Duke Math. J. 56(1988), 143–161. [21] M. Bestvina and M. Feighn, “Stable actions of groups on real trees,” Invent. Math. 121(1995), 287–321. J. W. Jones, “Hausdorff dimension and Kleinian groups,” Acta Math.