 Thus, mapping class groups—and even their finite-index subgroups—distinguish infinite-type surfaces. This answers Question 1. Corollary 1. Let S be an infinite-type surface.

The computational theory of Riemann–Hilbert problems (Lecture 4) by Thomas Trogdon

Our proof of Theorem 1. The first is an algebraic characterization of Dehn twists in terms of centralizers of elements see Section 4. The second ingredient comes from curve complexes. A multicurve is a finite set of distinct curves that admits representative embeddings with disjoint images. Our final theorem extends this result to infinite-type surfaces.

## Antiquariat Renner

Let us briefly establish some terminology for dealing with an infinite-type surface S. Note that domains are only defined up to isotopy and that each domain Y is itself a surface. A curve in S is essential in Y if its equivalence class contains an embedding that defines a curve in Y. Definition 2. Notice that Map S respectively acts on the sets of curves, multicurves, and domains of S. Lemma 2.

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By Theorem 2. Thus, f evidently preserves the orientation on Y and, consequently, all of S. The following fact will play a crucial role in our proof of Lemma 4. A domain Y as in Lemma 2. Since f fixes every curve disjoint from Y , we may use Theorem 2. In this section we prove Theorem 1. Lemma 3.

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• Farb, Margalit, A primer on mapping class groups - Стр 45;
• The Age of Innocence;
• Microfiltration and Ultrafiltration Membranes in Drinking Water.
• 3 editions of this work.

We now prove that isomorphisms of curve complexes are geometric. Proof of Theorem 1. The first step is to characterize finitely-supported elements. Definition 4. Assume f does not have finite support. Conversely, every finitely-supported mapping class may be written as a finite product of Dehn twists and half-twists see, e. As there are only countably many curves, it follows that Map S has only countably many finitely-supported elements.

## Complex analysis : an introduction to the theory of analytic functions of one complex variable

Therefore, when f has finite support, its conjugacy class in G is countable. Let f be an element of G. The following is a now consequence of Lemmas 2. Corollary 4. Reporting from:. Your name. Your email. Send Cancel. Check system status. Toggle navigation Menu.

• Global Knowledge Networks and International Development (Routledge Warwick Studies in Globalisation).
• Well Logging in Nontechnical Language?