Low Dimensional Topology

October 15, 2008

Mapping class groups of stabilized Heegaard splittings

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 5:11 pm

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo ambient isotopies that keep the Heegaard surface on itself.  A few things are known about mapping class groups of irreducible Heegaard splittings (in particular, those with high Hempel distance), but for stabilized Heegaard splittings, with one exception, essentially nothing is known.  (The exception is the genus two Heegaard splitting of the 3-sphere, about which almost everything is known.  But come on, genus two is easy, right?)

Recall that a Heegaard splitting is stabilized if it is the result of attaching a trivial handle to a lower genus splitting.  I’m really interested in the subgroup of the mapping class group coming from isotopy automorphisms of the ambient 3-manifold.  For the rest of the entry, this is what I will mean by the mapping class group.

The mapping class group of a stabilized Heegaard splitting should be related to the group of the lower genus splitting that it comes from, but it’s unclear how the two groups should be related.  In fact, I don’t think there is even a conjecture for what the mapping class group of a once-stabilized Heegaard splitting should be.  Well, to fix this absence, I’m going to state one.  It should probably be stated as a question rather than a conjecture, but conjectures are more dramatic (plus I think I can prove it for certain cases.)

First consider a once-stabilized Heegaard splitting of a surface cross an interval x I.  The initial Heegaard surface for S x I is a horizontal surface parallel to S that cuts the manifold into two surface cross interval pieces, each of which is a (trivial) compression body.  When we stabilize this splitting, we attach a one handle to each compression body.  The resulting Heegaard splitting has exactly one non-separating meridian disk on each side of the Heegaard surface, so any mapping class element must take these two meridian disks onto themselves.  This makes the mapping class group really easy to understand, and it turns out to be a semi-direcy product of the findamental group of S (coming from draggin the stabilization around S) and the infinite cyclic group (coming from spinning the stabilization around a reducing sphere).  (The proof of this is left as  an exercise for the reader.)

If S is a Heegaard surface then a stabilization of S is contained in a regular neighborhood of S and the above group is a subgroup of the stabilization’s mapping class group, which I’ll call the middle subgroup.  The stabilized surface can also be pushed into each of the handlebodies of the original Heegaard splitting, forming a stabilized Heegaard surface for this handlebody.  This Heegaard splitting of the handlebody has a unique meridian disk on one side (but not the other), making a description of its mapping class group not too hard (though I won’t describe it here).  This defines two more subgroups of the stabilized surface’s mapping class group, one for each of the original handlebodies, which I’ll call the left and right subgroups.  (There are probably better terms for these.  Leave your suggestions in the comments!)  The middle subgroup is contained in each of the left/right subgroups and the three groups generate a subgroup of the mapping class group of the stabilized splitting.  (I think it’s an amalgamated free product of the left and right subgroups.)

Finally, the original Heegaard splitting has its own mapping class group.  We can extend an automorphism of the original splitting to the stabilized surface by tweaking the original automorphism to be the identity in a disk neighborhood of the area where the stabilization is added.  Then it extends to the stabilized surface by making it the identity on the new handle.  Of course, there are lots of different ways that we can tweak the original map.  In fact, this construction is only defined up to composition by the middle subgroup.  If we look at all the ways of tweaking the original map we get a larger subgroup, which I’ll call the naive subgroup (again, feel free to suggest a better term) and there appears to be a short exact sequence from the middle subgroup into the naive subgroup into the original mapping class group.

Ok, so we now have four subgroups of the mapping class group of the stabilized splitting and they seem to have reasonably nice structures.  I’ll call the subgroup generated by all these subgroups the induced subgroup.  This subgroup is always persent in the mapping class group of the stabilized splitting.  But is it the whole group?  That’s the conjecture:

Conjecture: The induced subgroup (described above) should be the entire mapping class group of the stabilized Heegaard splitting.

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October 1, 2008

Mapping class groups and JSJ decompositions

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 10:48 am

Recall that a JSJ decomposition for a 3-manifold is a (minimal) embedded collection of tori that cut the 3-manifold into pieces such that each component is atoroidal or Seifert fibered.  Jaco-Shalen and Johannson (i.e. JSJ) showed that up to isotopy, every irreducible 3-manifold has exactly one such decomposition (which is empty, if the 3-manifold is already atoroidal or Seifert fibered).  If you’re also interested in Heegaard splittings of the 3-manifold in question (which I usually am) then you can consider a result of Tsuyoshi Kobayashi, that a strongly irreducible Heegaard splitting can always be isotoped to intersect a given incompressible torus in simple closed curves that are essential in both surface.

One nice application of this result, also due to Kobayashi, is a classification [1] of 3-manifolds with genus two Heegaard splittings and non-trivial JSJ decompositions.  In a recent preprint [2], Jungsoo Kim classifies finite group actions on these three manifolds that preserve both the Heegaard surface and the JSJ tori.  He considers each of three cases identified by Kobayashi and shows that in each case, the only possible (non-trivial) group actions are by the order two cyclic group or the order four dihedral group.  

These group actions are closely related to the mapping class groups of the Heegaard splittings (the group of automorphisms of the 3-manifold that take the Heegaard surface onto itself, modulo isotopies that preserve the Heegaard surface.)  In particular, the automorphisms of the 3-manifold induce a subgroup of the mapping class group of the Heegaard splitting.  In this case, I think the induced subgroup has to be isomorphic to the acting group, since a finte group cannot act on a genus two surface by isotopy trivial automorphisms.  (I don’t have a reference for this.)

What I’d like to know is: Is the whole mapping class group of the Heegaard splitting isomorphic to a maximal group acting on the 3-manifold in a way that preserves both the Heegaard surface and the JSJ surface.  In other words, can the mapping class group be realized by on of Kim’s group actions.  Hyam Rubinstein and I proved in [3] that every finite subgroup of the mapping class group of a Heegaard splitting can be realized by a group of automorphisms (essentially, a group action).  The problem is that an automorphism that preserves the Heegaard splitting may not preserve the JSJ tori.  Such an automorphism will take the JSJ surface to an isotopic surface, but this surface may intersect the Heegaard surface differently, so that there is not an isotopy preserving the Heegaard surface that takes the new JSJ tori to the originals.  At least, this is the problem when considering JSJ tori in general.  In this case, because the ways the JSJ tori can intersect a genus two surface are limited, and mostly understood, it seems like there might be a chance of showing that there’s only one way (up to isotopy) for the JSJ tori to intersect the Heegaard surface.  But I don’t understand Kobayashi’s classification well enough to figure it out, so for now it’s open.

[1] T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21 (1984), 437{455.

August 27, 2008

Involutions of handlebodies

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 2:42 pm

An involution of a topological space X is a homeomorphism h from X to itself such that h is not the identity but the composition of h with itself is the identity.  A simple involution of a surface can be constructed by drawing the standard picture of a surface, then skewering the surface with a horizontal line (as if you’re making a shish kabob) and applying a 180 degree rotation around the skewer.  If you think of that surface as being in 3-space then it bounds a handlebody and the involution extends to this handlebody as well.  The set of fixed points on the surface consists of 2g+2 isolated points (where g is the genus of the surface).  The set of fixed points in the handlebody consists of g+1 arcs connecting the isolated points in the surface (the intersection of the skewer with the handlebody).

If you do this for a genus two surface, the resulting involution has some really nice properties.  First, this is the only involution of a genus 2 surface with exactly six fixed points.  (Note: an earlier version of this sentence was incorrect, but Saul sent me an e-mail pointing that out.)  Second, this involution sends each simple closed curve in the surface onto itself (up to isotopy), which implies that the involution extends to every handlebody bounded by the surface.  (It also means that this involution induces the identity map on the curve complex!)  If the genus two surface is a Heegaard splitting, then it bounds two handlebodies and the involution extends to both of them, i.e. to the ambient 3-manifold.  The quotient of each handlebody by this involution is a ball and the fixed point set descends to boundary parallel arcs in these balls.  In this way, every 3-manifold with a genus two Heegaard splitting can be written as a double branched cover of a three bridge knot in the 3-sphere, and this is one of the main links between the theory of Heegaard splittings and the theory of bridge positions of knots.  (See my previous post on branched covers and bridge positions.)

For higher genus handlebodies, there are more types of involutions than the one above.  However, a nice classification of them is given in a recent preprint by Pantaleoni and Piergallini [1].  They show that the involutions fall into two groups.  First, there are fixed point free involutions, which occur only in odd genus surfaces and look like you’ve taken a standard picture of the handlebody, skewered it through the middle hole (so the skewer misses the surface) and then rotated it 180 degrees.  (The paper has a nice picture of this which I won’t reproduce here.)  The second type of involution, one with fixed points, breaks down into some pieces where there are arcs of fixed points, some pieces with loops of fixed points, and some free pieces.  There is a very nice picture of this in the paper, which I won’t reproduce here, either.  The proof is nice and short and is based on finding a system of disks that behaves nicely under the involution.

The reason I’m writing about this here is, of course, because it’s related to mapping class groups of Heegaard splittings.  If there is an involution of a 3-manifold that takes a Heegaard surface to itself then this involution either swaps the two handlebodies or it extends to an involution of the two handlebodies.  In the case when the involution extends, that means that the involution of the surface commutes with the gluing map between the handlebodies.  It might be possible to use this fact to classify Heegaard splittings that admit such involutions, though that seems like a hard problem.

I should also mention that you can think of an involution as a group action by Z_2 on the space.  McCullough, Miller and Zimmerman [2] have proved some things about general group actions on handlebodies and Zimmerman has used ideas along these lines to proved results about group actions on hyperbolic 3-manifolds.

[2] D. McCullough, A. Miller and B. Zimmermann, Group actions on handlebodies, Proc. London Math. Soc. 59 (1989), 373-416.

August 12, 2008

Mapping class groups of incompressible surfaces

Filed under: 3-manifolds,Mapping class groups — Jesse Johnson @ 4:21 pm

I’ve mentioned in previous posts the idea of the mapping class group of a Heegaard surface: the group of automorphisms of the ambient 3-manifold that take the Heegaard surface to itself, modulo isotopies that leave the surface on itself.  There’s also the subgroup of mapping class elements that are isotopy trivial as automorphisms of the ambient manifold.  This group is the kernel of the induced map from the mapping class group of the Heegaard surface to the mapping class group of the ambient manifold.

There’s nothing in this definition that requires that the surface be a Heegaard surface; we can define the same group for any embedded surface, in particular for incompressible surfaces.  For a separating incompressible surface, elements of this group come from automorphisms of the complementary components that induce matching automorphisms of their shared boundaries.  For non-separating surfaces, non-trivial automorphisms come from automorphism of the complement that match up when the two boundary components making the incompressible surface are glued together.

As with Heegaard surfaces, one can ask whether non-trivial elements of incompressible surface’s mapping class group can be isotopy trivial in the ambient manifold.  For Heegaard surfaces, there are lots of examples, but for incompressible surfaces, the only obvious examples are the leaves of surface bundles.  Pushing a leaf around the monodromy brings it back to itself by a non-trivial automorphism, so this mapping class element is in the kernel.

The disparity in the number of examples comes from the fact that Heegaard surfaces are in many ways more floppy than incompressible surfaces.  In particular, the map from the fundamental group of a Heegaard surface to the fundamental group of the ambient manifold is nowhere close to injective, while the same map for an incompressible surface is injective.  Thus an automorphism of an incompressible surface coming from an isotopy of the manifold implies an element of the fundamental group of the 3-manifold such that conjugating the surface’s fundamental group by this element brings it back to itself.   (You can find this element by following the base point in the surface during the isotopy.)  In other words, the normalizer of the surface’s fundamental group, as a subgroup of the 3-manifold’s fundamental group, is non-trivial.

I wouldn’t be at all surprised if it turns out that the only incompressible surfaces with non-trivial automorphism that are trivial in the ambient manifold are leaves of surface bundles.  In fact, it seems like the algebraic situation, a surface subgroup with non-trivial normalizer, should only happen in a surface bundle.  However, my knowledge of results along these lines is very limited.  Do any readers know if this is the case?  Also, if you know of any paper where mapping class groups of incompressible surfaces are discussed, please put that in a comment too.

July 3, 2008

Mapping classes vs. Homeomorphisms

Filed under: Mapping class groups — Jesse Johnson @ 2:24 pm

Though not quite as exciting as a possible proof of the Riemann hypothesis, a paper on realizing the mapping class group of a surface as a subgroup of the group of self-homeomorphisms [1] caught my eye a couple of days ago. Recall that the set of all homeomorphisms from a surface S to itself form a group that I’ll call Aut(S) or the automorphism group. The mapping class group of S, which I’ll write Mod(S), is the quotient of this group by isotopies of the surface. The quotient construction implies a homomorphism from Aut(S) onto Mod(S). The paper above proves that for a surface of genus at least two, there is no reverse homomorphism from Mod(S) into Aut(S) such that composing the maps produces the identity on Mod(S).

I had though this was already known, but apparently it was only known for genus at least 5 (or 3 if you replace Aut(S) with the group Diff(S) of diffeomorphisms.) The introduction to the paper lists the previously known results. You can ask a similar question for subgroups of Mod(S) as well, i.e. whether there’s a map from the subgroup into Aut(S) or Diff(S) that composes to the identity on that subgroup. For infinite cyclic groups, the answer is an almost immediate yes (just pick any representative for a generator). For finite subgroups, the answer is a much harder to prove yes; this is the Nielsen Realization Theorem (which was proved by Steve Kerckhoff, not by Nielsen). I wonder which infinite, non-cyclic subgroups of Mod(S) can be realized as groups of homeomorphisms?

The proof in the paper examines a certain relation that is discussed in Farb and Margalit’s primer on mapping class groups: Given a separating loop in a surface, you can write a Dehn twist around that loop as a composition of Dehn twists along loops in the interior of one of the complementary components or along loops in the other complementary component. Since both give you a Dehn twist along the same loop, these give you a relation. The authors then use the machinery defined in [2] (where the “no” answer for genus at least 5 is proved) to show that such a relation cannot exist in Aut(S).

[2] V. Markovic, Realization of the mapping class group by homeomorphisms. Inventiones Mathematicae
168 , no. 3, 523–566 (2007)

April 3, 2008

Follow up to More on mapping class groups

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 8:55 am

I now have an answer to the question on mapping class groups of Heegaard splitting I asked a few weeks ago. It appears that for any Heegaard splitting of the 3-torus, the exact sequence from the kernel to the MCG of the Heegaard splitting to the MCG of the 3-manifold does not split (i.e. there’s no homomorphism back from the MCG of the 3-manifold). In fact, there is no injection from the mapping class group of the 3-torus to and surface mapping class group by a Theorem of Farb and Masur [1]. (Thanks go to Yair Minsky for telling me where to find the result.) Specifically, they prove that any homomorphism from an irreducible lattice in a semi-simple Lie group of rank at least two into a mapping class group has finite image. The mapping class group of the 3-torus is SL(3,Z), which is of this form, so such a homomorphism cannot be an injection.

March 13, 2008

Group theory, as told by a Heegaard splitter, Part 3

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 11:54 am

Ok, one more post about group theory and then I’ll drop it. Ben Webster asked a question on my last mapping class group post about whether you could think about mapping class groups of Heegaard splittings in terms of actions on the fundamental group of the 3-manifold. Well it happens that almost any question about Heegaard splittings can be studied in terms of fundamental groups without even knowing what a Heegaard splitting is, though you need to look at more than just the fundamental group of the 3-manifold.

Stallings [1] introduced the idea of a splitting homomorphism as follows: Consider two homomorphisms from a genus g surface fundamental group onto two rank g free groups. Consider the product of the kernels of these two homomorphisms in the surface group. The product is a normal subgroup so we can take the quotient G of the surface group by this product. There are induced homomorphisms from the two free groups into G, making a commutative diamond.

Jaco showed [2] that for any two such homomorphisms, there is a way of identifying a genus g surface with the boundaries of two handlebodies, forming a 3-manifold and a Heegaard splitting for that 3-manifold, such that the inclusion maps between the fundamental groups are the two homomorphisms we started with. It follows that G is the fundamental group of the 3-manifold. Given two splitting homomorphisms (i.e. commutative diamonds of this form), we can consider a collection of four homomorphisms between corresponding elements forming a commutative cube. Such a cube corresponds to a homeomorphism between the two 3-manifolds that takes one Heegaard splitting to the other. Thus two splitting homomorphisms form a commutative cube if and only if they correspond to homeomorphic Heegaard splittings of homeomorphic 3-manifolds.

One might want to determine whether two Heegaard splittings are isotopic rather than just homeomorphic. In this case, one can consider two splitting diagrams that share the same bottom group (i.e. the fundamental group of the 3-manifold). This is sort of a commutative pair of glasses. If there are homomorphisms between the three remaining pairs of corresponding groups that make a commutative diagram (a squished cube?) then the corresponding Heegaard splittings are (in most cases) isotopic. (By “most cases” I mean for the large class of 3-manifolds for which an isomorphism between fundamental groups corresponds to a unique homeomorphism.)

So this gives one a completely algebraic way to study homeomorphism classes and isotopy classes of Heegaard splittings. The mapping class group of a Heegaard splitting is the set of commutative cubes between a fixed splitting homomorphism and itself. (Multiplication and inverses are easy to work out.) Jaco describes in [3] a construction on the commutative diamond that corresponds to stabilizing the Heegaard splitting.

Now that I’ve described all this, I should point out that it’s probably not a very useful approach to Heegaard splittings. As far as I know there are only four or five papers in existence on splitting diagrams, and though things have been proved about them, nothing has been proved using them. But who knows, maybe someone who knows more about group theory than the average Heegaard splitter could find something new.

[1] Stallings, John. How not to prove the Poincare conjecture. Topology Seminar, Wisconsin, 1965, Ann. of Math. Studies, No. 60 (Princeton 1966).

[2] Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379.

[3] Jaco, William. Stable equivalence of splitting homomorphisms. 1970 Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) pp. 153–156 Markham, Chicago, Ill.

February 21, 2008

Open Books and Heegaard Splittings

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 1:05 pm

I’m not sure exactly the best way to pose an open problem about open book decompositions and Heegaard splittings, but here’s the idea: Recall that an open book decomposition for a 3-manifold is a link in the 3-manifold and a surface bundle structure on the link complement such that each leaf of the bundle is a spanning surface for the link. The closure of the union of any two leafs form a Heegaard surface. The problem I want to suggest is, given a Heegaard splitting, to understand the set of open book decompositions that induce a Heegaard splitting in the same isotopy class. (Note that a Heegaard splitting induced by an open book has Hempel distance at most two, so it’s easy to find splittings that aren’t induced by any open books.)

The most immediate reason to want to understand this set of open book decompositions is that spinning around the circle factor of the surface bundle determines an automorphism of the Heegaard splitting (and as you may have noticed I’m currently rather interested in automorphisms of Heegaard splittings.) So understanding the collection of open book decompositions would tell us something about Heegaard splitting’s the mapping class group.

But there’s another reason that the connection between open books and Heegaard splittings is interesting. There is a very strong, and reasonably well understood connection between open books and contact structures, due to the fact that the plane field defined by an open book decomposition can be deformed into a canonical contact structure. So open book decompositions may be a good way to find some relation ships between contact structures and Heegaard splittings. (For example one might ask if the existence of a tight contact structure implies the existence of a strongly irreducible Heegaard splitting or vice versa.)

The way that open books interact with contact structures and Heegaard splittings is slightly different. Given an open book decomposition, there is a construction called Hopf plumbing that essentially consists of connect summing the open book decomposition with a copy of a Hopf link in the 3-sphere. This decreases the Euler characteristic of the leaf by one and changes the number of components of the link by one. It affects the induced Heegaard splitting by adding a stabilization. The effect on the contact structure depends on whether the hopf link has positive or negative linking number. I think the way it works (though someone correct me if I’m getting this wrong) is that a positive Hopf plumbing has no effect on the contact structure, while a negative plumbing will turn a tight contact structure into an over-twisted one.

Giroux and Goodman [1] have used the correspondence between open books and contact structures, and the fact that overtwisted contact structures are determined entirely by their homology class, to show that open book decompositions that determine the same homology class in a given 3-manifold are stably equivalent under Hopf plumbing. They note that this is related to stable equivalence for Heegaard splittings. Trying to prove stable equivalence of open books via their connection to Heegaard splittings turns out to be more difficult because it’s unclear how to calculate the homology class of the open book from its spine in the Heegaard surface. (I vaguely remember seeing a preprint related to this, but I wasn’t able to find it when I went back and looked.) But the point is, there seems to be something there to explore.

February 15, 2008

More on mapping class groups

Filed under: 3-manifolds,Heegaard splittings,Mapping class groups — Jesse Johnson @ 12:20 pm

I’m running out of open questions to write about, so I want to urge all you readers to submit your own open questions via the comments box on the open problems page. For now, though, here’s a problem about mapping class groups of Heegaard splittings that may not be that hard, but I haven’t had time to think about it myself.

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient manifold that take the Heegaard splitting onto itself, modulo isotopies that keep the Heegaard surface on itself. (This is sometimes referred to as the Goeritz group of the Heegaard splitting, though other times the Goeritz group refers specifically to Heegaard splittings of the 3-sphere.) Since every automorpism of the Heegaard splitting comes from an automorphism of the ambient manifold, there is a canonical homomorphism from the mapping class group of the Heegaard splitting to the mapping class group of the ambient manifold. In general, this homomorphism won’t be one-to-one or onto, though by stabilizing the Heegaard splitting, we can find a splitting for any manifold such that the homomorphism is onto.

Once one has a Heegaard splitting for which this homomorphism is onto, one can ask further whether there is an isomorphic copy of the mapping class group of the 3-manifold contained in the mapping class group of the Heegaard splitting. More generally, given a Heegaard splitting, consider the short exact sequence from the kernel of the canonical homomorphism to the mapping class group of the Heegaard splitting to the mapping class group of the image of the homomorphism in the mapping class group of the 3-manifold. Does this short exact sequence split? (I.e. is there an isomorphism from the mapping class group of the 3-manifold back to the mapping class group of the Heegaard splitting such that the composition of this isomorphism with the canonical homomorphism is the identity?)

If the short exact sequence splits then the mapping class group of the Heegaard splitting is a semi-direct product of the kernel of the canonical homomorphism and its image (which in many cases will be the mapping class group of the 3-manifold). We can also ask the weaker question: Given a 3-manifold, is there always a Heegaard splitting such that the short exact sequence splits? And an even weaker version: can every mapping class group of a 3-manifold be embedded in the mapping class group of a surface? I don’t know if an answer to this last question is known, though it seems like it might be easier to answer through other means. If one can find a 3-manifold whose mapping class group is not a subgroup of a surface mapping class group then no Heegaard splitting of this 3-manifold can have a split short exact sequence.

November 20, 2007

Quasi-isometries of the curve complex

Filed under: Curve complexes,Mapping class groups — Jesse Johnson @ 4:20 pm

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.

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