Proceedings of the Second ISAAC Congress Volume 2: This Project Has Been Executed with Grant No. 11-56 from the Commemorative Association for the Japan World Exposition (1970) |
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Editor:
| Begehr, Heinrich G. W. Gilbert, R. P. Kajiwara, Joji |
Series title: | International Society for Analysis, Applications, and Computation Ser. |
ISBN: | 978-1-4613-7971-3 |
Publication Date: | Sep 2011 |
Publisher: | Springer
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Book Format: | Paperback |
List Price: | AUD $765.95 |
Book Description:
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Let 8 be a Riemann surface of analytically finite type (9, n) with 29 2+n> O. Take two pointsP1, P2 E 8, and set 8 ,1>2= 8 \ {{P1' P2}}. Let PI Homeo+(8;P1,P2) be the group of all orientation preserving homeomor phismsw: 8 -+ 8 fixingP1, P2 and isotopic to the identity on 8. Denote byHomeot(8;Pb P2) the set of all elements ofHomeo+(8;P1, P2) iso topic to the identity on 8 ,P2' ThenHomeot(8;P1,P2) is a normal sub pl group ofHomeo+(8;P1,P2). We setIsot(8;P1,P2) =Homeo+(8;P1,P2)/...
More DescriptionLet 8 be a Riemann surface of analytically finite type (9, n) with 29 2+n> O. Take two pointsP1, P2 E 8, and set 8 ,1>2= 8 \ {{P1' P2}}. Let PI Homeo+(8;P1,P2) be the group of all orientation preserving homeomor phismsw: 8 -+ 8 fixingP1, P2 and isotopic to the identity on 8. Denote byHomeot(8;Pb P2) the set of all elements ofHomeo+(8;P1, P2) iso topic to the identity on 8 ,P2' ThenHomeot(8;P1,P2) is a normal sub pl group ofHomeo+(8;P1,P2). We setIsot(8;P1,P2) =Homeo+(8;P1,P2)/ Homeot(8;p1, P2). The purpose of this note is to announce a result on the Nielsen Thurston-Bers type classification of an element [w] ofIsot+(8;P1,P2). We give a necessary and sufficient condition for thetypeto be hyperbolic. The condition is described in terms of properties of the pure braid [b ] w induced by [w]. Proofs will appear elsewhere. The problem considered in this note and the form ofthe solution are suggested by Kra's beautiful theorem in [6], where he treats self-maps of Riemann surfaces with one specified point. 2 TheclassificationduetoBers Let us recall the classification of elements of the mapping class group due to Bers (see Bers [1]). LetT(R) be the Teichmiiller space of a Riemann surfaceR, andMod(R) be the Teichmtiller modular group of R. Note that an orientation preserving homeomorphism w: R -+ R induces canonically an element (w) EMod(R). Denote by&.r(R)(middot;,.) the Teichmiiller distance onT(R). For an elementXEMod(R), we define a(x)= inf &.r(R)(r,x(r)).