Proceedings of the Second ISAAC Congress

Angebote / Angebote:

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)/ 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)(·, .) the Teichmiiller distance onT(R). For an elementXEMod(R), we define a(x)= inf &.r(R)(r, x(r)).