Convexity and Aigner’s Conjectures

Aigner [1] proposed three conjectures to better understand this order.

1. (1)

   (The fixed numerator conjecture) Let $p,q$ and $i$ be positive integers such that $i>0,p<q$ and $gcd(p,q)=1,gcd(p,q+i)=1$ then

   $$m_{p/q}<m_{p/(q+i)}.$$

2. (2)

   (The fixed denominator conjecture) Let $p,q$ and $i$ be positive integers such that $i>0,p<q$ and $gcd(p,q)=1,gcd(p+i,q)=1$ then

   $$m_{p/q}<m_{(p+i)/q}.$$

3. (3)

   (The fixed sum conjecture) Let $p,q$ and $i$ be positive integers such that $i>0,p<q$ and $gcd(p,q)=1,gcd(p-i,q+i)=1$ then

   $$m_{p/q}<m_{(p-i)/(q+i)}.$$

The fixed numerator conjecture was proved in [8] using ideas from cluster algebra theory and snake graphs. Their proof relies on a connection between Christoffel paths and Markov numbers (see [2] for a discussion) which provides a formulation of Markov numbers as continuant polynomials. The proof is rather technical and introduces several interesting tools for studying paths in the Cayley graph of the monoid $\mathbb{N}^{2}$ with the usual generators. In [9], the authors prove the fixed numerator conjecture by using ”transformations” on paths in the Cayley graph. We will show how these results can be deduced by considerations of convexity established in [7]. Our proof is represented visually in Figure 4.

The question of counting for Markov triples, that is for $R>0$ estimating the size of the set of Markov triples $(x,y,z)$ for which the sup norm is less than $R$, was first investigated by Gurwood in his thesis. He established an asymptotic formula using a correspondence between Markov and Farey trees. Somewhat later Zagier [10] obtained an improved error term. At the time of writing the best result is due to a Rivin and the author [7]:

$$M(R)=C(\log R)^{2}+O(\log R\log\log R),$$

for a numerical constant which has a geometric interpretation.

The method introduced in [7] differs from that of Gurwood and Zagier in that it relies on ideas of H. Cohn to interpret the question in terms of geometry of hyperbolic surfaces. In [4] Cohn shows that the Markov triple $(1,1,1)$ is, up to multiplication by 3, the image under the character map $\chi$ of the modular torus. Recall that the modular torus is the quotient of the upper half $\mathbb{H}$ plane by $\Gamma$, the commutator subgroup of $\text{PSL}(2,\mathbb{Z})$, acting by Mobius transformations. Secondly, he shows that the permutations and the Vieta flips used to construct Markov’s binary tree are induced by automorphisms of the fundamental group of the torus and concludes that $\text{Aut}(\mathbb{Z}*\mathbb{Z})$ acts transitively on Markov triples. It is well known that if $\rho:\mathbb{Z}*\mathbb{Z}\rightarrow\text{PSL}(2,\mathbb{R})$ is a discrete faithful representation and $\gamma$ a closed curve homotopic on the surface $X=\mathbb{H}/\rho(\mathbb{Z}*\mathbb{Z})$, then $\ell_{X}(\gamma)$, the length of the closed geodesic homotopic to $\gamma$ on the surface $X$, can be computed from the trace using the relation

$$2\cosh\ell_{X}(\gamma)/2=|\text{tr\,}\rho(\gamma)|.$$

We note that the absolute value of the trace of an element of $\text{PSL}(2,\mathbb{R})$ is well defined. Now identifying $\mathbb{Z}*\mathbb{Z}$ with the fundamental group of the modular torus Cohn shows that if $x$ is a Markov number then there exists a simple loop $\gamma$ such that

$$x=\frac{1}{3}|\text{tr\,}\rho(\gamma)|.$$

so there is an explicit relation between Markov numbers and lengths of simple geodesics on the modular torus. In [7] the simple closed curves are embedded as a subset of primitive elements in the integer lattice $\mathbb{Z}^{2}\subset\mathbb{R}^{2}$ and the length function $\gamma\mapsto\ell_{X}(\gamma)$ is shown, using a simple geometric argument, to extend to a proper convex function $\ell:\mathbb{R}^{2}\mapsto\mathbb{R}_{+}$. Thus the counting problem for Markov numbers can be settled by counting integer points in the set

$$\ell^{-1}([0,R])=R\ell^{-1}([0,1])\subset\mathbb{R}^{2}$$

It is easy to visualize this set using a very short computer program see for example Figure 1.

In summary, given a fraction $p/q$ one obtains a primitive lattice point $(q,p)\in\mathbb{Z}^{2}$ and so a closed simple geodesic $\gamma_{p/q}\subset X$ such that

$$M_{p/q}=\frac{2}{3}\cosh\left(\frac{\ell_{X}(\gamma_{p/q}))}{2}\right)=\frac{2}{3}\cosh(\|(q,p)\|_{s}),$$

where $\|.\|_{s}$ is the norm induced by the function $\frac{1}{2}\ell:\mathbb{R}^{2}\mapsto\mathbb{R}_{+}$.

Thus the fixed numerator conjecture is equivalent to : Let $p,q$ and $i$ be positive integers such that $i>0,p<q$ and $p$ is coprime with both $q,q+i$ then

$$\|(q,p)\|_{s}<\|(q+i,p)\|_{s}.$$

It is natural now to ask oneself whether we need to restrict to integers. In fact, using quite simple geometric arguments one can prove a result, in a non discrete setting, which yields Aigner’s conjectures as a corollary:

We give a brief account of the geometry and topology of the once punctured torus in Sections 2 and 3. This is followed in Section 4 by the proof of the convexity of the length function $\ell$ which appeared in [7]. We conclude with the proof of Theorem 1.1 in Section 5.

One could of course state many more theorems using these ideas but our purpose here is simply to show why Aigner’s conjectures are manifestations of the convexity of the length function.

Finally we note that it is unlikely that purely geometric methods would yield a proof of Frobenius’ conjecture as a natural variation of it is false [6].

I thank Vlad Segesciu and Louis Funar for many useful conversations over the years concerning this subject. I am grateful too to Xu Binbin for comments on the preliminary versions of the text.

It has been known since the time of Fricke that, as a consequence of Riemann’s Uniformization Theorem, the moduli space of hyperbolic structures on the three holed sphere and the one holed torus can be identified with semi algebraic subsets of $\mathbb{R}^{3}$. More precisely, one obtains a hyperbolic structure $X$ on $\Sigma$ from a discrete faithful representation $\rho$ of $\pi_{1}(\Sigma)$ into $\mathrm{isom}(\mathbb{H})$ such that $\mathbb{H}/\rho(\pi_{1}(\Sigma))$ is homeomorphic to $\Sigma$. One can lift any such $\rho$ from a representation into $\mathrm{isom}(\mathbb{H})^{+}$, which is isomorphic to $\text{P}\mathrm{SL}(2,\mathbb{R})$, to a representation

$$\tilde{\rho}:\pi_{1}(\Sigma)\rightarrow\text{SL}(2,\mathbb{R}).$$

Doing this allows one to consider the trace $\text{tr\,}\tilde{\rho}(\gamma)\in\mathbb{R}$ for any element $\gamma$ of the free group so that many problems can be reduced to questions of linear algebra. In particular one defines a character map

$$\chi:X\mapsto(\text{tr\,}\tilde{\rho}(\alpha),\text{tr\,}\tilde{\rho}(\beta),\text{tr\,}\tilde{\rho}(\alpha\beta)),$$

which provides an embedding of the moduli space of hyperbolic structures on $\Sigma$. Goldman [5] studied the dynamical system defined by the action of the group of outer automorphisms of $\pi_{1}(\Sigma)\simeq\mathbb{Z}*\mathbb{Z}$ on the moduli space. If $\phi$ is an automorphism then it acts on the representations

$$\tilde{\rho}\mapsto\tilde{\rho}\circ\phi^{-1}$$

and evidently this induces an action on the points in the image of the embedding above. It turns out that, by applying Cayley-Hamilton theorem to $\mathrm{SL}(2,\mathbb{R})$, it is easy to show that this action is the restriction of polynomial diffeomorphisms of $\mathbb{R}^{3}$.

An important aspect of the theory of the action of the group of outer automorphisms is that it admits an invariant function. In the case of the pair of orientable surfaces this is

$$\kappa:(x,y,z)\mapsto x^{2}+y^{2}+z^{2}-xyz.$$

The level set $\kappa^{-1}(0)$ has five connected components - the singleton $(0,0,0)$ and four codimension one manifolds homeomorphic to $\mathbb{R}^{2}$. The non trivial components can be identified with the Teichmueller space of a once punctured torus. It is well known that there are integer points in $\kappa^{-1}(0)\cap\mathbb{R}_{+}^{3}$, for example $(3,3,3)$, and that they form a single orbit for the action of the outer automorphisms. From such a point one obtains a Markov triple, that is a solution in positive integers of the Markov cubic

$$x^{2}+y^{2}+z^{2}-3xyz=0$$

simply by dividing through by $3$. This map is natural in that it is a conjugation of automorphisms of the level set $\kappa^{-1}(0)$ with the automorphisms of the Markov cubic. So, since we will not be interested in arithemetic properties of these solutions, we will also refer in what follows to elements of $\kappa^{-1}(0)\cap\mathbb{N}^{3}$ as Markov numbers. An integer which appears in a Markov triple is called a Markov number.

An element $\gamma\in\mathbb{Z}*\mathbb{Z}$ is called primitive, if there exists an automorphism $\phi$, such that $\gamma=\phi(\alpha)$. If $\phi(\delta)=\beta$, then $\gamma$ and $\delta$ are called associated primitives.

Following [7] let $\phi:\mathbb{Z}*\mathbb{Z}\rightarrow\mathbb{Z}^{2}$ be the canonical abelianizing homomorphism. The kernel of $\phi$ is a characteristic subgroup, that is it is $\text{Aut}(\mathbb{Z}*\mathbb{Z})$-invariant, so there is a homorphism from $\text{Aut}(\mathbb{Z}*\mathbb{Z})$ to the automorphisms of $\mathbb{Z}^{2}$ namely $\mathrm{GL}(2,\mathbb{Z})$. In fact this homomorphism is surjective and faithful. Moreover,

1. (1)

   If $\gamma$ is primitive then $\phi(\gamma)$ is a primitive element of $\mathbb{Z}^{2}$.

2. (2)

   $\text{Aut}(\mathbb{Z}*\mathbb{Z})$ acts transitively on primitive elements of $\mathbb{Z}*\mathbb{Z}$

3. (3)

   $\mathrm{GL}(2,\mathbb{Z})$ acts transitively on primitive elements of $\mathbb{Z}^{2}$.

It is useful, and this is essentially Cohn’s point of view [4], to identify $\mathbb{Z}*\mathbb{Z}$ with the fundamental group of the holed torus and think of $\mathrm{Out}(\mathbb{Z}*\mathbb{Z})$, the outer automorphism group of $\mathbb{Z}*\mathbb{Z}$, as being the mapping class group of the holed torus. The outer automorphism group is generated by the so-called Nielsen transformations, which either permute the basis $\alpha,\beta$, or transform it into $\alpha\beta,\beta$, or $\alpha\beta^{-1},\beta$. Both these latter transformations can be realized topologically as Dehn twists on the holed torus. We identify $\mathbb{Z}^{2}$ with the first homology of the holed torus $\Sigma$ and will call the cover $\tilde{\Sigma}\rightarrow\Sigma$ corresponding to $\ker\phi$ the homology cover or maximal abelian cover of $\Sigma$.