Unstable Quasi - Geodesics in Teichm

In this note we will construct an example of a quasi-geodesic in the Te-ichm uller space of a surface with the following instability property: Theorem For any hyperbolic surface S whose Teichm uller space T (S) has dimension at least 4, there is a bi-innnite quasi-geodesic L in T (S) such that 1. L projects to a compact part of the moduli space M(S), but 2. L does not lie in a bounded neighborhood of any geodesic. (Recall that a quasi-geodesic, or (K;)-quasi-geodesic for K > 1; > 0, is a path for which the lengthànd distance d between endpoints of any subsegment satisfy`Kd + .) This theorem answers in the negative a question asked us by M. Mitra (see 7] for a related sharper question), and should be contrasted with Theorem 4.2 from 6], which gives the following stability property: Given a xed compact set C M(S) and K > 1; > 0, there is a constant R so that, if G is any geodesic whose projection to M(S) lies in C, and if L is any (K;)-quasi-geodesic joining two points of G, then L lies in an R-neighborhood of G. This stability property holds, by comparison, in hyperbolic spaces for all geodesics G, but in Teichm uller space it fails if G does not project to a compact set. This suggests that \in the thick part of Teichm uller space things are essentially hyperbolic". Our theorem, on the other hand, shows that the roles of the geodesic and the quasi-geodesic cannot be reversed, even in the thick part, and hence that this stability is somewhat delicate. Note of course that when dim(T (S)) = 2 the theorem is false: T (S) is then the hyperbolic plane, in which all quasi-geodesics are uniformly close to geodesics. We refer the reader to 4] for basic notions of Teichm uller theory; to 2] and 3] for material on Thurston's measured lamination (or foliation) spaces and on pseudo-Anosov homeomorphisms; and to 1] for more on quasi-geodesics and their stability properties in hyperbolic spaces. Proof Fix any essential proper subdomain Y of S, other than an annulus or a thrice-punctured sphere. By \essential" we mean that 1 (Y) injects into 1 (S). Let h : S ! S be a homeomorphism which preserves Y , is the identity outside Y , and whose restriction to Y is in a pseudo-Anosov mapping class.