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.
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.
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.
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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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.
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Name of resource. Problem URL. Describe the connection issue. SearchWorks Catalog Stanford Libraries. Riemann surfaces and algebraic curves : a first course in Hurwitz theory. Language English. Text in English. Series London Mathematical Society student texts ; Online Available online. Cambridge Core Full view. Science Library Li and Ma. Ahlfors, E. Calabi, M. Morse, L. Sario, and D. Spencer, Editors pp.
Regularity of mappings inverse to Sobolev mappings
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Note from archivist cs. Calculus and Analysis Complex Analysis Conformal Mapping Ahlfors Five Island Theorem Let f z be a transcendental meromorphic function , and let be five simply connected domains in with disjoint closures Ahlfors Functions, Meromorphic. MathComp database - Short view of documents To display full information of a single document, click on the eye.