A lot has been published in the last few years about the curve complex. This is a simplicial complex determined by a given surface, whose vertices are isotopy classes of simple closed curves in the surfaces and faces bounding sets of pairwise disjoint, non-parallel loops. Most of the motivation for studying it comes from its connections to mapping class groups, Teichmuller space and, recently, the ending laminations conjecture. It also makes good fodder for Gromov/coarse geometry techniques since its local structure is untenable. Of course, my interest in the curve complex is motivated by the fact that it’s very useful for understanding Heegaard splittings. But for this blog entry I want to talk about a result that may very well have no relevance to Heegaard splittings:
Kasra Rafi and Saul Schleimer recently posted a very nice, relatively short preprint [1] proving a number of results about quasi-isometries of curve complexes. A quasi-isometric map between two geometric objects is a map such that for any two points in the domain, the distance between their images is bounded above and below by a linear function, plus or minus a constant, of their distance in the original space. A quasi-isometry is a quasi-isometric map whose image is k-dense for some k. (In other words, every point in the range must be within distance k of the image of a point from the domain.) By allowing this sort of flexibility, one essentially ignores the local behavior of the map in favor of its large scale or asymptotic properties.
Every automorphism of a surface induces an isometry of its curve complex, and it is known [2] that except in a few trivial cases, there is a one-to-one correspondence between isometries of the curve complex and automorphisms of the surface. The isometries are completely determined by their large scale behavior, in the sense that the only isometry that moves each point a bounded distance is the identity isometry. (This is true for any finite bound.) Rafi and Schleimer have shown that for most surfaces (those whose curve complex has a connected Gromov boundary) every quasi-isometry of the curve complex is a bounded distance from an actual isometry.
It is a reasonably simple exercise to show that two curve complexes are isometric if and only if their underlying surfaces are homeomorphic. Rafi and Schleimer’s result imples that two curve complexes are quasi-isometric if and only if their underlying surfaces are homeomorphic. The proof uses a result of Ursula Hamenstaedt [2], which states that for any Cayley graph of the mapping class group of a surface, every quasi-isometry is a bounded distance from a true isometry.
