Proceedings of the Second ISAAC Congress

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 81>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 Kras 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)).