{"product_id":"9780792365983","title":"Proceedings of the Second ISAAC Congress","description":"Let 8 be a Riemann surface of analytically finite type (9, n) with 29 - 2+n\u0026gt; O. Take two pointsP1, P2 E 8, and set 8 ,1\u0026gt;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\u0026amp;.r(R)(*,.) the Teichmiiller distance onT(R). For an elementXEMod(R), we define a(x)= inf \u0026amp;.r(R)(r,x(r)).","brand":"Heinrich G.W. Begehr, R.P. Gilbert, Joji Kajiwara","offers":[{"title":"Default Title","offer_id":45857904984382,"sku":"9780792365983","price":6087.67,"currency_code":"ZAR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4357\/7662\/products\/9780792365983.jpg?v=1707326949","url":"https:\/\/exclusivebooks.co.za\/products\/9780792365983","provider":"Exclusive Books Online","version":"1.0","type":"link"}