Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the question, if legitimate at all, should have been restricted to interesting manifestations of a hyperbolic-parabolic-elliptic subdivision, then I can fully agree (although part of the idea was to interpret the question as you see fit); I left it open ended primarily because of the Weil trichotomy, which is of completely different kind and is so much more than a hierarchy, and in relation to which I was interested in hearing other people's opinions and elaborations. See, for instance, how Edward Frenkel, in a fascinating Bourbaki talk, builds upon the Weil trichotomy to introduce a parallel between Langlands and electro-magnetic dualities, which he uses as a springboard for the ideas from physics that have entered the arena of the geometric Langlands program. Or take the cherished three-sided parallel between the basic three-dimensional (from the point of view of etale cohomology) objects and their branched coverings: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathrm{Spec}(\mathbb{Z})$, and $S^3$, with primes in number fields corresponding to knots in threefolds, $\log{p}$ corresponding to hyperbolic length, etcetera.

To those who were not convinced that there was a neat trichotomy of algebraic surfaces (arguing they should instead form a tetrachotomy by Kodaira dimension), let alone in higher dimensional algebraic geometry, I refer to Sándor Kovács's answer here, which demonstrates rather eloquently the fundamental trichotomy of birational geometry:

How "frequent" are smooth projective varieties with (anti-)ample canonical bundle?

Original post. For many purposes, notably in classification hierarchies or in Weil's "big picture" of the fundamental unity in mathematics, it seems as if mathematical reality is more accurately captured by trichotomies than by two-sided dictionaries or questions of "either/or." The most basic is of course the trichotomy negative - zero - positve embodied by the complete ordered field $(\mathbb{R},<)$ --- this is the Arrow of Time, if you will, or the conditioning of a dynamical system into states of past/present/future. As evidenced by some of the examples below, this trichotomy underlies varied, if crude, classification schemes in mathematics.

Other trichotomies arise from closer examinations of a mathematical parallel. Mathematicians have always been fond of discovery by analogy; they take very seriously the intuitions supplied by different yet loosely connected fields. In doing so, they are guided by a tacit, platonic belief in the fundamental Unity of mathematics. An example is the similarity between finite geometries and Riemann surfaces. To explain this parallel, indeed to make sense of it, it is necessary to provide a "middle column" in the dictionary: the arithmetic geometry of number fields and arithmetic surfaces. This leads to the trichotomy that Weil explained so lucidly in a letter (which he wrote in 1940 in prison for his refusal to serve in the army) to his sister, the philosopher Simone Weil. This point of view led, as we know, to an entire new field of mathematical inquiry.

Below I have listed some other cherished mathematical trichotomies. I am interested in seeing yet others, perhaps more specialized. This is my question: add more trichotomies to the list. Furthermore, I am interested in any reflections anyone might have, such as pertaining, for instance, to any of the following questions. Is 3 the most ubiquitous number in coarse classification schemes? Is it fair to say that a given trichotomy echoes the primeval trichotomy $(-,0,+)$? In a given trichotomy, is there a natural "middle column" of a corresponding three-sided dictionary? Is this "middle column" in any way the most fundamental, the most interesting, or the most elusive?

Trichotomies in mathematics: some examples.

• The fabric of topology, geometry, and analysis is the real line $\mathbb{R}$. Tarski's eight axioms characterize it in terms of a complete binary total order <, a binary operation +, and a constant 1. (Multiplication comes afterwards - it is implied by Tarski's axioms - and so does the Bourbaki definition of the reals as the complete ordered field). The sign trichotomies $(<,=,>)$ and $(-,0,+)$ ensuing from those axioms have repercussions throughout all of mathematics.

• For example, there are three constant curvature spaces, leading to the three maximally symmetric geometries: hyperbolic, flat (or Euclidean), and elliptic (e.g. spherical forms).

• Locally symmetric spaces fall into three types: non-compact type, flat, and compact type.

• In complex analysis, there are three simply connected cloths: the Riemann surfaces $\Delta$, $\mathbb{C}$, and $\hat{\mathbb{C}}$.

• The connected component of the group of conformal automorphisms of a compact Riemann surface is one of the following three: trivial, $S^1 \times S^1$, $\mathrm{PGL}_2(\mathbb{C})$.

• The complexity of fundamental groups, as showcased first of all by topological surfaces: genuinely non-abelian (perhaps we could say: anabelian) - abelian (or more generally, containing a finite index nilpotent subgroup) - and trivial (or more generally, finite). This is of course related to the subject of growth of finitely generated groups, brought forward by Lee Mosher's answer.

• In dynamics, a fixed point (or a periodic cycle) can be either repelling, indifferent, or attracting.

• In Thurston's work on surface homeomorphisms, elements of the mapping class group are classified according to dynamics into three types: pseudo-Anosov, reducible, and finite-order.

• In algebraic geometry, the positivity of the canonical bundle is central to the classification and minimal model problems. More generally, positivity is a salient feature of algebraic geometry. For a delightful discussion, see Kollar's review of Lazarsfeld's book "Positivity in algebraic geometry." (Bull. AMS, vol. 43, no. 2, pp. 279-284). The most basic example is the trichotomy of algebraic curves (rational, elliptic, general type).

• In birational algebraic geometry, at a very coarse level, there are three kinds of varieties out of which a general variety is made: rational curves, Calabi-Yau manifolds, and varieties of general type (or hyperbolic type, if you prefer). For example, an algebraic surface either: 1) admits a pencil of rational curves; or 2) admits a pencil of elliptic curves or is abelian or K3 (or a double quotient of a K3); or else 3) it is of general type. Abelian and K3 are examples of Calabi-Yau manifolds.

• More concretely, consider smooth hypersurfaces $X \subset \mathbb{P}^n$. They divide into three types, according to how their degree $d$ compares with the dimension. If $d \leq n$, they contain plenty of rational curves (certainly uncountably many). If $d = n+1$, they are an example of a Calabi-Yau manifold, and typically contain a countably infinite number of rational curves. (The generating function of the number of rational curves of a given degree is then a very interesting function, of significance in the physics of quantum gravity.) And if $d \geq n+2$, then $X$ is of general type, and it is conjectured to typically contain only finitely many rational curves. (More precisely, Bombieri and Lang have conjectured that a variety of general type contains only finitely many maximal subvarieties not of general type).

• In diophantine geometry, rational points are supposed to come from rational curves and abelian varieties. The sporadic examples are believed to be finitely many. This leads to the following trichotomy for the growth rate of the number of rational points of bounded (big, i.e. exponential) height: polynomial growth - logarithmic growth - $O(1)$. Furthermore, even in dimension 1, it is for abelian varieties that the situation is the deepest and the most mysterious.

• In topology, it seems as if the interesting dimensions fall into three qualitatively different ranges: $d = 3$, $d = 4$, and $d \geq 5$. (Although this might be stretching it a bit too much). Of these, four dimensions -- the "middle column" -- is the most mysterious, and also the most relevant for physics.

• The "Weil trichotomy," of course, goes at least as far back to Kronecker and Dedekind: curves over $\mathbb{F}_q$ - number fields - Riemann surfaces. Class field theory and Iwasawa theory are particularly eloquent examples of this trichotomy. Another example is of course the zeta function and the Riemann hypothesis.

• One would be tempted to extend the latter trichotomy to [non-Archimedean world ($p$-adic, profinite) - global arithmetic - Archimedean world (geometry, topology, complex variables)], if the middle column did not subsume (much of) the flanking columns. Likewise the triple [$l$-adic cohomology-motive-Hodge structure] would probably not be admissible. Here is a variation on the theme (you may find it to be rubbish, in which case throw it away). There are two ways of completing (or taking limits of) the regular polygons $C_n$. The first is to think of $C_n$ as $\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ and take the direct limit (in this case, union, or synthesis: $\rightarrow$), which is $\mathbb{Q}/\mathbb{Z}$. Completing, we get the circle $S^1 = \mathbb{R}/\mathbb{Z}$, which is the simplest manifold. The second is to think of $C_n$ as $\mathbb{Z}/n$ and take the projective limit (or deconstruction: $\leftarrow$), which is $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$, the profinite version of the circle. In this way, Archimedean (continuous) objects and $p$-adic objects may be seen as the two possible different limits (synthesis and deconstruction) of the same finite objects. Taking $C_n$ to be more general finite groups, we get essentially all the Lie groups, on the one hand; and all the profinite groups, on the other hand.

• That we live in three perceptible spatial dimensions does not, of course, fit our bill. But in 1984, Manin published an article ("New dimensions in geometry") in which, guided by ideas from number theory (Arakelov geometry) and physics (supersymmetry), he proposed that there are three kinds of geometric dimensions, modeled on the affine superscheme $\mathrm{Spec} \mathbb{Z}[x_i;\xi_j]$, an "object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}/2$-graded supercommutative rings." Here, $\xi_j$ are "odd," anticommuting variables, commuting with the "even" variables $x_i$. See the three coordinate axes $x, \xi$ and $\mathrm{Spec} \mathbb{Z}$ in his picture of "three-space-2000." The arithmetic axis $\mathrm{Spec} \mathbb{Z}$ is implicit in complex algebraic geometry, and is essential in problems such as the Ax-Grothendieck theorem and the construction of rational curves in Fano manifolds.

• In the theory of linear groups there is, loosely speaking, a trichotomy: $\mathbb{G}_m$ (linear tori) - semisimple - $\mathbb{G}_a$ (unipotent).

• Algebraic groups: reductive - abelian variety - unipotent. Especially, the classification of one-dimensional groups: $\mathbb{G}_m$ - $E$ - $\mathbb{G}_a$. (Thanks, Terry Tao!)

• Variant: among commutative algebraic groups, there are: multiplicative type - abelian varieties - additive type (unipotent).

• $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ are the only finite-dimensional associative division algebras over the continuum. (Thanks Paul Reynolds, Teo B, and Sam Lewallen!)

• The most basic PDEs of physics: the wave equation (hyperbolic) - the heat and Schrodinger equations (parabolic) - the Laplace equation (elliptic). (Thanks Alexandre Eremenko!)

• An infinite finitely-generated group has $1,2$ or $\infty$ ends. (Thanks shane.orourke and Artie Prendergast-Smith!)

• A random walk is either transient, null recurrent, or positive recurrent. (Thanks Vaughn Climenhaga!)

• Zeta functions can by dynamical (Artin-Mazur); arithmetical on schemes of finite type over $\mathbb{Z}$ (Riemann and Hasse-Weil); and geometric (Selberg's zeta function of a hyperbolic surface).

• In Model theory, there is an important trichotomy between super-stable theories, strict-stable (stable but not superstable) theories, and non stable theories.

• It seems fair to say that there are three kinds of three-dimensional simply connected spaces: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathbb{Spec}(\mathbb{Z})$ compactified at archimedean infinity, and $S^3$. This brings about the Mazur knotty dictionary and the fruitful analogy between primes and knots (especially hyperbolic knots).

• A finitely-generated infinite group has 1, 2, or infinitely many ends. (But if you leave out the word "infinite", it becomes a tetrachotomy; this seems to be a weakness of the question.)
– user5117
Feb 2, 2013 at 21:14
• From the movie $\pi$: "Hold on. You have to slow down. You're losing it. You have to take a breath. Listen to yourself. You're connecting a computer bug I had with a computer bug you might have had and some religious hogwash. You want to find the number 216 in the world, you will be able to find it everywhere. 216 steps from a mere street corner to your front door. 216 seconds you spend riding on the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere." Feb 3, 2013 at 10:05
• This feels like an awfully arbitrary and contentless question to me. This seems like asking, what are some interesting sets with 3 elements? Voting to close. A more reasonable question would be to ask for examples where there is an interesting hyperbolic-parabolic-elliptic trichotomy and connections between these examples. Feb 3, 2013 at 13:28
• Let's please not be quick to close this question. There are some intriguing clusters of answers, some along a (negative, 0, positive) axis, and others along a (closed, open, closed) axis [if I may put it crudely], and others along other axes still. There is a certain amount of erudition in both the question and answers, and there is some potential to glimpse "analogies between analogies" in the words of Ulam. It could be a useful exercise to pursue this. Feb 4, 2013 at 2:29
• Meta thread here: tea.mathoverflow.net/discussion/1525/…. Please upvote this comment so it appears "above the fold" Feb 4, 2013 at 16:36

The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: the $sphere$; the connected sum of $g$ $tori$, ; the connected sum of $k$ $real$ $projective$ $planes$. this is a simple example of the trichotomy.sphere can be taken as $0$ tori. so $SPHERE$ serves the middle column.
Doing Bernoulli percolation (each edge is kept with probability $p$, erased otherwise, independently) on a transitive graph yields 0, 1 or infinitely many infinite clusters, and each of these behaviours may occur.
More generally, consider a transitive (connected locally finite) graph, i.e. consider a (connected locally finite) graph where all vertices play the same role: formally, we assume that the automorphism group of the graph acts transitively on the vertices. Consider a random subgraph of this graph: you are allowed to randomly erase vertices and edges. Assume that your random process is invariant under the symmetries of your graph: for any automorphism $f$, the distribution of your random subgraph is the same as that of your $f$-translated subgraph. Finally, assume that your process is insertion-tolerant: if $x$ is a vertex or an edge, and if $E$ is an event which does not look at $x$ and which has positive probability, then there is a positive probability that $E$ occurs and $x$ is not erased. Then, almost surely, the number of infinite connected components belongs to $\{0,1,\infty\}$.
(In the case of the so-called Bernoulli percolation, this gives rise to the critical parameters $p_c$ and $p_u$. Another interesting trichotomy in statistical mechanics is subcritical/critical/supercritical; this trichotomy has already been mentioned in this thread, in the setting of PDEs.)