By D. B. A. Epstein
Quantity 2 is split into 3 elements: the 1st 'Surfaces' includes a piece of writing by means of Thurston on earthquakes and by way of Penner on traintracks. the second one half is entitled 'Knots and 3-Manifolds' and the ultimate half 'Kleinian Groups'.
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Additional resources for Low-dimensional Topology and Kleinian Groups: Warwick and Durham 1984
Sample text
These conditions are also necessary. 2 it is clear that two sequences with nonempty intersection are equivalent if and only if they have the same intersection. A sequence x with nonempty intersection D(a, r) (possibly r = 0) cannot be equivalent to any sequence y = {D(ai , ri )} with empty intersection, since for any i with a ∈ / D(ai , ri ) we have [T − ai ]x = [T − ai ]D(a,r) = |a − ai | > ri , whereas [T − ai ]y ≤ [T − ai ]D(ai ,ri ) = ri . Finally, two sequences x = {D(ai , ri )} and y = {D(aj , rj )} with empty intersection which are not mutually cofinal cannot be equivalent.
If i D(ai , ri ) is nonempty, then the intersection is either a disc D(a, r) of positive radius, in which case (ζa,r , ζGauss ] extends in a natural way to the closed path [ζa,r , ζGauss ], or a point a ∈ D(0, 1), in which case the half-open path (ζa,r , ζGauss ] extends to a closed path by adding on the corresponding type I point a as an endpoint. Similarly, if i D(ai , ri ) = ∅, we must adjoin a new endpoint of type IV in order to close up (ζa,r , ζGauss ]. It is clear that cofinal sequences define the same half-open path, so the point closing up this path depends only on the corresponding seminorm [ ]x and not on the sequence defining it.
We will usually write ζGauss instead of ζ0,1 . We will call two nested sequences of closed discs equivalent if they define the same point in D(0, 1). 3. Two nested sequences of closed discs {D(ai , ri )}, {D(aj , rj )} are equivalent if and only if (A) each has nonempty intersection, and their intersections are the same, or (B) both have empty intersection, and each sequence is cofinal in the other. ) Proof. 2) is a decreasing one, it is clear that two sequences {D(ai , ri )} and {D(aj , rj )} are equivalent if either condition (A) or (B) is satisfied.