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I am in Kyoto right now, attending the twenty-first Nevanlinna colloquium (update: took a while to write this post – now I’m in Sydney for the Clay lectures). Yesterday, Junjiro Noguchi gave a plenary talk on Nevanlinna theory in higher dimensions and related Diophantine problems. The talk was quite technical, and I did not understand it very well; however, he said a few suggestive things early on which struck a chord.

The talk started quite accessibly, being concerned with the fundamental equation

$a +b = c$

where $a,b,c$ are coprime positive integers. The abc conjecture, formulated by Oesterlé and Masser, says that for any positive real number $\epsilon$, there is a constant $C_\epsilon$ so that

$\max(a,b,c) \le C_\epsilon\text{rad}(abc)^{1+\epsilon}$

where $\text{rad}(abc)$ is the product of the distinct primes appearing in the product $abc$. Informally, this conjecture says that for triples $a,b,c$ satisfying the fundamental equation, the numbers $a,b,c$ are not divisible by “too high” powers of a prime. The abc conjecture is known to imply many interesting number theoretic statements, including (famously) Fermat’s Last Theorem (for sufficiently large exponents), and Roth’s theorem on diophantine approximation (as observed by Bombieri).

Roth’s theorem is the following statement:

Theorem(Roth, 1955): Let $\alpha$ be a real algebraic number. Then for any $\epsilon>0$, the inequality $|\alpha - p/q| < q^{-(2+\epsilon)}$ has only finitely many solutions in coprime integers $p,q$.

This inequality is best possible, in the sense that every irrational number can be approximated by infinitely many rationals $p/q$ to within $1/2q^2$. In fact, the rationals appearing in the continued fraction approximation to $\alpha$ have this property. There is a very short and illuminating geometric proof of this fact.

In the plane, construct a circle packing with a circle of radius $1/2q^2$ with center $p/q,1/2q^2$ for each coprime pair $p,q$ of integers.

This circle packing nests down on the $x$-axis, and any vertical line (with irrational $x$-co-ordinate) intersects infinitely many circles. If the $x$ co-ordinate of a vertical line is $\alpha$, every circle the line intersects gives a rational $p/q$ which approximates $\alpha$ to within $1/2q^2$. qed.

On the other hand, consider the corresponding collection of circles with radius $1/2q^{2+\epsilon}$. Some “space” appears between neighboring circles, and they no longer pack tightly (the following picture shows $\epsilon = 0.2$).

The total cross-sectional width of these circles, restricted to pairs $p/q$ in the interval $[0,1)$, can be estimated as follows. Each $p/q$ contributes a width of $1/2q^{2+\epsilon}$. Ignoring the coprime condition, there are $q$ fractions of the form $p/q$ in the interval $[0,1)$, so the total width is less than $\frac 1 2 \sum_q q^{-1-\epsilon}$ which converges for positive $\epsilon$. In other words, the total cross-sectional width of all circles is finite. It follows that almost every vertical line intersects only finitely many circles.

Some vertical lines do, in fact, intersect infinitely many circles; i.e. some real numbers are approximated by infinitely many rationals to better than quadratic accuracy; for example, a Liouville number like $\sum_{n=1}^\infty 10^{-n!}$.

Some special cases of Roth’s theorem are much easier than others. For instance, it is very easy to give a proof when $\alpha$ is a quadratic irrational; i.e. an element of $\mathbb{Q}(\sqrt{d})$ for some integer $d$. Quadratic irrationals are characterized by the fact that their continued fraction expansions are eventually periodic. One can think of this geometrically as follows. The group $\text{PSL}(2,\mathbb{Z})$ acts on the upper half-plane, which we think of now as the complex numbers with non-negative imaginary part, by fractional linear transformations $z \to (az+b)/(cz+d)$. The quotient is a hyperbolic triangle orbifold, with a cusp. A vertical line in the plane ending at a point $\alpha$ on the $x$-axis projects to a geodesic ray in the triangle orbifold. A rational number $p/q$ approximating $\alpha$ to within $1/2q^2$ is detected by the geodesic entering a horoball centered at the cusp. If $\alpha$ is a quadratic irrational, the corresponding geodesic ray eventually winds around a periodic geodesic (this is the periodicity of the continued fraction expansion), so it never gets too deep into the cusp, and the rational approximations to $\alpha$ never get better than $C/2q^2$ for some constant $C$ depending on $\alpha$, as required. A different vertical line intersecting the $x$-axis at some $\beta$ corresponds to a different geodesic ray; the existence of good rational approximations to $\beta$ corresponds to the condition that the corresponding geodesic goes deeper and deeper into the cusp infinitely often at a definite rate (i.e. at a distance which is at least some fixed (fractional) power of time). A “random” geodesic on a cusped hyperbolic surface takes time $n$ to go distance $\log{n}$ out the cusp (this is a kind of equidistribution fact – the thickness of the cusp goes to zero like $e^{-t}$, so if one chooses a sequence of points in a hyperbolic surface at random with respect to the uniform (area) measure, it takes about $n$ points to find one that is distance $\log{n}$ out the cusp). If one expects that every geodesic ray corresponding to an algebraic number looks like a “typical” random geodesic, one would conjecture (and in fact, Lang did conjecture) that there are only finitely many $p/q$ for which $|p/q - \alpha| < q^{-2}(\log{q})^{-1-\epsilon}$ for any $\epsilon > 0$.

A slightly different (though related) geometric way to see the periodicity of the continued fraction expansion of a quadratic irrational is to use diophantine geometry. This is best illustrated with an example. Consider the golden number $\alpha = (1+\sqrt{5})/2$. The matrix $A=\left( \begin{smallmatrix} 2 & 1 \ 1 & 1 \end{smallmatrix} \right)$ has $\left( \begin{smallmatrix} \alpha \ 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} \bar{\alpha} \ 1 \end{smallmatrix} \right)$ as eigenvectors (here $\bar{\alpha}$ denotes the “conjugate” $1-\alpha$), and thus preserves a “wedge” in $\mathbb{R}^2$ bounded by lines with slopes $\alpha$ and $\bar{\alpha}$. The set of integer lattice points in this wedge is permuted by $A$, and therefore so is the boundary of the convex hull of this set (the sail of the cone). Lattice points on the sail correspond to rational approximations to the boundary slopes; the fact that $A$ permutes this set corresponds to the periodicity of the continued fraction expansion of $\alpha$ (and certifies the fact that $\alpha$ cannot be approximated better than quadratically by rational numbers).

There is an analogue of this construction in higher dimensions: let $A$ be an $n\times n$ integer matrix whose eigenvalues are all real, positive, irrational and distinct. A collection of $n$ suitable eigenvectors spans a polyhedral cone which is invariant under $A$. The  convex hull of the set of integer lattice points in this cone is a polyhedron, and the vertices of this polyhedron (the vertices on the sail) are  the “best” integral approximations to the eigenvectors. In fact, there is a $\mathbb{Z}^{n-1}$ subgroup of $\text{SL}(n,\mathbb{Z})$ consisting of matrices with the same set of eigenvectors (this is a consequence of Dirichlet’s theorem on the structure of the group of units in the integers in a number field). Hence there is a group that acts discretely and co-compactly on the vertices of the sail, and one gets a priori estimates on how well the eigenvectors can be approximated by integral vectors. It is interesting to ask whether one can give a proof of Roth’s theorem along these lines, at least for algebraic numbers in totally real fields, but I don’t know the answer.

I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic $3$-manifold $M$ contains a closed $\pi_1$-injective surface. Equivalently, $\pi_1(M)$ contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing question in $3$-manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic $3$-manifold groups were LERF, one would be able to deduce that all hyperbolic $3$-manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic $3$-manifold groups are LERF; if successful, this would therefore resolve some of the most important outstanding problems in $3$-manifold topology (in fact, I would say: the most important outstanding problems, by a substantial margin).

In fact, the argument appears to work for hyperbolic manifolds of every dimension $\ge 3$, and possibly more generally still. Details on the argument of Markovic-Kahn are scarce (Vlad informs me that they expect to have a preprint in a few weeks) but the sketch of the argument presented by Kahn is compelling. Roughly speaking, the argument (as summarized by Ian Agol in a comment at Jesse’s blog) takes the following form:

1. Given $M$, for a sufficiently big constant $R$, one can find “many” immersed, almost totally-geodesic pairs of pants (i.e. thrice-punctured spheres) with geodesic boundary components (i.e. “cuffs”) of length very close to $2R$. In fact, one can further insist that the complex length of the boundary geodesic is very close to $2R$ (i.e. holonomy transport around this geodesic does not rotate the normal bundle very much).
2. Conversely, given any geodesic of complex length very close to $2R$, one can find many such pairs of pants that it bounds, and moreover one can find them so that the normal to the geodesic pointing in to the surface is prescribed.
3. If one takes a sufficiently big collection of such geodesic pairs of pants, one has enough of them in oppositely-aligned pairs along each boundary component, that they can be matched up (by some version of Hall’s marriage theorem), and furthermore, matched up with a definite prescribed “twist” along the boundary components
4. One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is $\pi_1$-injective

Many aspects of this argument have a lot in common with some previous attempts on the surface subgroup conjecture, including one recent approach by Bowen (note: Bowen’s approach is known to have some fatal difficulties; the “twist” in 3. above specifically addresses some of them). All of these points deserve some comments.

First, where do the pairs of pants come from? If $P$ is a totally geodesic pair of pants with boundary components of length close to $2R$, the pants $P$ retract onto a geodesic spine, i.e. an immersed totally geodesic theta graph, whose edges all have length close to $2R$, and which meet at angles very close to $120$ degrees. One can cut this spine up into two pieces, which are obtained by exponentiating the edges of an infinitesimal (almost)-planar tripod for length $R$.

Given a tripod $T$ in some plane in the tangent space at some point of $M$, one can exponentiate the edges for length $R$ to construct such a half-spine; if $T$ and $T'$ are a pair of tripods for which the exponentiated endpoints nearly match up, with almost opposite tangent vectors, then the resulting half-spines can be glued up to make a spine, and thickened to make a pair of pants. One key idea is to use the exponential mixing property of the geodesic flow on a hyperbolic manifold, e.g. as proved by Pollicott. Given some tolerance $\epsilon$, once $R$ is sufficiently large, the mixing result shows that the set of such pairs of tripods for which such a matching occurs have a definite density in the space of all pairs (and in fact, are more and more equidistributed in this space, in probability). In fact, one may even insist that two of the pairs of prongs join up to make some specific closed geodesic of length almost $2R$, and vary the pair of third prongs a very small amount so that they glue up. This takes care of the first two points; this seems quite uncontroversial (exponential mixing comes in, I suspect, to know that one doesn’t need to wiggle the pair of third prongs much, having paired the first two pairs).

The matching (i.e. the gluing up of opposite pant cuffs) apparently is done by some variant of Hall’s marriage theorem. One needs to know (I think) that for any finite set of cuffs to be glued, the set of other cuffs that they could potentially be glued to is at least as big in cardinality. This probably needs some thought, but it is plausibly true: given a cuff, it can be glued to any cuff which is almost oppositely aligned to it, and since there is some tolerance in the angle of gluing — this is where dimension at least $3$ is necessary — and moreover, since oriented cuffs are almost equidistributed, one can always find “more” cuffs that are opposite, up to a bit of tolerance, to any given subset of cuffs (of course, more details are necessary here). There is an extra wrinkle to the argument, which is that the gluing must be done with a “twist” of a definite amount, so that cuffs are not glued up in such a way that the perpendicular geodesic arcs joining pairs of cuffs match up.

(Update 8/8: I think there must necessarily be more details to the matching argument, as very loosely described above. There are at least two additional issues that must be dealt with in order to perform a matching: a parity issue (since each pants has an odd number of cuffs) and a homology issue (if the argument relativizes, so that one fixes some collection of cuffs in advance and glues up everything else, one concludes a posteriori that the union of the unglued cuffs is homologically inessential). Probably the parity issue (and more subtle divisibility issues) can be solved by gluing with real-valued weights, then approximating a real solution by a rational solution, and multiplying through to clear denominators. Maybe the homology issue does not arise, if in fact the argument doesn’t relativize.) Both these issues suggest that one does not specify in advance a collection of pants to be glued up, but rather wants to glue up a definite number of pants from some subset.)

This issue of a twist is important for the 4th point, which is perhaps the most delicate. In order to know that the resulting surface is $\pi_1$-injective, one must use geometry. A closed (immersed) surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is $\pi_1$-injective. One way to see this is to observe that a geodesic loop in the surface is almost geodesic in the manifold; the ambient negative curvature means that the geodesic can be shrunk (by the negative of the gradient of length in the space of loops) to become geodesic in the ambient manifold; if it is close to being geodesic at the start, it very quickly becomes totally geodesic, without getting much shorter. Any closed geodesic in a hyperbolic manifold is essential.

If one builds a surface by gluing up almost totally geodesic pieces in such a way that there is almost no angle along the gluing, the resulting surface is almost geodesic, and therefore injective. However, one must be very careful to control the geometry of the pieces that are glued, and this is hard to do if the injectivity radius is very small. A geodesic pair of pants has area $2\pi$ no matter how long its boundary components are. So if the boundary components have length $2R$, then at the points where they are thinnest, they are only $e^{-R}$ across. If cuffs are glued where the pants are thinnest, even if the gluing angle is very small, the surfaces themselves might twist through a big angle in a very short time. So one needs to make sure that the thinnest part of one pants are glued up to a thicker part of the next, which is glued to a thicker part of the next . . . and so on. This is the point of introducing the twist before gluing: the twists accumulate, and before one has glued $R$ pieces together, one has entered the thick part of some pants, where the injectivity radius is bounded below by some universal constant.

Anyway, this seems like a really spectacular development, with an excellent chance of working out. Some of the ingredients — e.g. the exponential mixing of the geodesic flow — work just as well in variable negative curvature. In fact, some version of it should work for arbitrary hyperbolic groups (using Mineyev’s flow space). Without knowing more details of the argument, one can’t say how delicate the last part of the argument is, and how far it generalizes (but readers are invited to speculate . . .)