Trichotomies in mathematics

Question

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 étale cohomology) objects and their branched coverings: $\mathbb{P}_{\mathbb{F}_q}^1$, $\operatorname{Spec}(\mathbb{Z})$, 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 - positive 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, $S^1 \times S^1$, $\operatorname{PGL}_2(\mathbb{C})$.

• 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 Kollár'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 $\operatorname{Spec} \mathbb{Z}[x_i;\xi_j]$, 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 $\operatorname{Spec} \mathbb{Z}$ in his picture of "three-space-2000." The arithmetic axis $\operatorname{Spec} \mathbb{Z}$ 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 be 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: $\mathbb{P}_{\mathbb{F}_q}^1$, $\operatorname{Spec}(\mathbb{Z})$ 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).

Answer 1

Almost everything in mathematics, indeed, can be "hyperbolic", "parabolic" or "elliptic". Like PDE's, Riemann surfaces, or manifolds of higher dimension, fractional-linear transformations, fixed points of a map, etc.

Not even mentioning the 3 kinds of the conic sections:-)

Of course this can be traced back to the fundamental trichotomy "positive", "zero" and "negative".

In differential geometry we have three great areas: "positive curvature", "negative curvature" and "zero curvature".

See also:

Strichartz, Robert S. Realms of mathematics: elliptic, hyperbolic, parabolic, sub-elliptic. Math. Intelligencer 9 (1987), no. 3, 56–64.

Answer 2

Each set in a topological space partitions the space into three parts: interior, boundary, exterior.

Answer 3

Finite, countable, and uncountable.

Those are the three distinctions we have for cardinalities. For most people uncountable would usually mean $2^{\aleph_0}$. But even for set theorists, given a model of ZFC, the finite sets are finite, the countable sets are countable, and the rest is madness$^*$.

Replacing cardinality by topological-measure theoretic properties of subsets of an ordinal $\kappa$, there are non-stationary sets (small); stationary sets (big, but not too big); and clubs (which is practically everything).

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$^*$ Good madness! Like a smile without a cat.

Answer 4

There are exactly three finite-dimensional, associative division algebras over $\mathbb R $: $$\mathbb R, ~~ \mathbb C,~~\mathbb H$$

This was also mentioned in Paul Reynolds' answer.

Arnol'd thought this was a particularly fundamental trichotomy, and he has a fascinating table of "related" trichotomies in his book "Arnol'd's Problems" (sorry for the apostrophe catastrophe). Here's a picture I uploaded a few years ago:

Arnol'd's table

I wonder if these triples are in the same "family" as $(-1 ,0 , 1) $. At first they seem to be different because there is no distinguished element, analogous to $0$ - but perhaps $\mathbb C$ plays this role? I think the proof that there are three division algebras might actually use the $(-1,0,1)$ trichotomy, but I don't have a moment to think about it right now - maybe someone can leave a comment confirming this? (Edit: I could also imagine $\mathbb R$ playing the role of $0$, because it is the identity element for the tensor product, for example - any opinions either way?)

Another question I find interesting, which I'd hoped to think about at some point for fun, asks not just for (related) occurrences of 3-element sets in mathematical classifications, but more generally, for related occurrences of $n$-element sets for small $n$. I'd always dreamed of taking the number $5$ of regular platonic solids, for example, and trying to deduce from this as many other "small, finite number classifications" as possible.

Answer 5

Every system of linear equations over the reals say has either no solution, or one solution, or infinitely many solutions.

Answer 6

The representation type of a finite dimensional algebra is either finite, tame or wild. This is a theorem of Drozd.

Answer 7

1. After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence of positive real numbers either (a) converges to a positive real; (b) diverges to $+\infty$; or (c) diverges to $0$. In nonstandard analysis, these trichotomies become those of being bounded, negative unbounded, or positive unbounded, or of being infinitesimal, unbounded, or neither (i.e. both bounded, and bounded away from zero). These are of course variants of the basic $(-,0,+)$ trichotomy.

2. Up to isomorphism, there are only three types of (connected) one-dimensional algebraic groups over an algebraically closed field: the additive group of the field, the multiplicative group of that field, and the elliptic curves over that field. (This last family would definitely be the "middle column".) This is of course connected to many of the other trichotomies previously mentioned. On the Riemann surface side, it comes from the fact that all one-dimensional connected complex groups are isomorphic to ${\bf C}/\Gamma$ for some discrete subgroup $\Gamma$ of ${\bf C}$, which can have rank 0 (additive case), 1 (multiplicative case), or 2 (elliptic curve case).

3. If one squints at it in just the right way, the classification of finite simple groups is a trichotomy: cyclic, Lie type (including Lie over F_1, i.e. alternating group), or sporadic. (Of course, it can be sliced in many other ways; counting the items in this Wikipedia list, for instance, would make it a tetratetracontachotomy.) Sometimes it is conceptually useful to split up the large Lie type groups into three regimes: large characteristic but bounded rank; large rank but bounded characteristic (including the alternating groups); and large characteristic and large rank. (Alternatively, one can partition into bounded rank, alternating, and unbounded rank.) One can debate as to which of these categories is the "middle column".

4. If $\xi_1,\xi_2,\xi_3$ are three frequencies with $\xi_3 = \xi_1+\xi_2$, then we have the Littlewood-Paley trichotomy: (a) "high-low" interactions with $|\xi_1| \gg |\xi_2|$ and $|\xi_1| \sim |\xi_3|$; (b) "low-high" interactions with $|\xi_1| \ll |\xi_2|$ and $|\xi_2| \sim |\xi_3|$; and (c) "high-high" interactions with $|\xi_1| \sim |\xi_2|$ and $|\xi_1| \gg |\xi_3|$. (One has to carefully demarcate the boundaries between these three possibilities to ensure it is a true trichotomy.) To an algebraic geometer, this would reflect the Y-shaped nature of the amoeba of the set $\{ (\xi_1,\xi_2,\xi_3): \xi_3 = \xi_1 + \xi_2 \}$. This trichotomy is important in harmonic analysis and PDE, and in particular in the paradifferential calculus of products and paraproducts (see e.g. this blog post of mine). Often, one of the three interactions will be the most dominant, reflecting either a high-to-low frequency cascade or a low-to-high frequency cascade, but it depends heavily on the situation. Note that this trichotomy is basically a variant of the $(<,=,>)$ trichotomy.

5. (ADDED LATER) Another variant of the $(<,=,>)$ trichotomy: most basic examples of semilinear PDE (or more precisely, a semilinear PDE problem, such as an initial value problem in a certain function space) can be classified as subcritical, critical, or supercritical, depending on whether the nonlinear component of the PDE is "weaker than", "comparable to", or "stronger than" the linear component in a suitable asymptotic limit (usually the fine scale/high frequency limit, although for scattering theory the coarse scale/low frequency limit is the relevant one instead). This distinction (which can usually be made precise through a scaling analysis or dimensional analysis) is often decisive in determining the difficulty level of the PDE problem. For instance, the regularity problem for 3D Navier-Stokes is supercritical and thus considered close to intractable, but 2D Navier-Stokes is critical and was solved decades ago. The global analysis of Ricci flow (with surgery) was considered supercritical until Perelman discovered new monotone quantities that made it critical, which was absolutely necessary for Perelman to be able to execute the rest of Hamilton's program and solve the Poincare and geometrisation conjectures. In this trichotomy, the critical (or scale-invariant) case is generally viewed as the most interesting and delicate, with some very nice mathematical tools coming into play to control the interaction between different scales. Perhaps it should also be pointed out that this trichotomy is orthogonal to the elliptic/parabolic/hyperbolic trichotomy, which only concerns the linear component of the PDE and not the nonlinear component, and all nine combinations (critical elliptic PDE, supercritical parabolic PDE, etc.) are studied in the literature.

6. (ADDED YET LATER) In analysis, there are basically three scenarios that prevent a weakly convergent sequence $f_n$ of functions in some function space from being strongly convergent in that space: (a) escape to "horizontal infinity" (basically, the support of the function runs off to spatial infinity, i.e. moving bump type examples); (b) escape to "vertical infinity" (the peaks of the function go to infinity, e.g. a sequence of approximations to the identity converging weakly but not strongly to a delta function); and (c) escape to "frequency infinity" (the functions become increasingly oscillatory). If one can shut down all three modes of escape then one can recover strong convergence, and thus also strong (pre)compactness, cf. the Arzela-Ascoli theorem which has three hypotheses (compact domain, pointwise boundedness, equicontinuity) to shut down (a), (b), and (c) respectively. In Section 2.9 of Lieb-Loss, these three scenarios are called "wanders off to infinity", "goes up the spout", and "oscillates to death" respectively.

Answer 8

A simple yet useful one: An irreducible complex representation of a compact Lie group is either 'real', 'quaternionic', or 'complex'. That is, it is the complexification of a real irreducible, or it can be considered quaternionic through the existence of an equivariant conjugate-linear (real-)automorphism $j$ that squares to $-I$, or it is neither.

The statement combines Schur's Lemma and the fact that there are three associative real division algebras, here seen through complex eyes.

Answer 9

In algebraic topology, the sequence $$ \text{ordinary cohomolgy},\qquad K\text{-theory},\qquad \text{elliptic cohomology} $$ closely parallels the trichotomy in the classification of algebraic groups ($\mathbb G_a$, $\mathbb G_m$, elliptic curves).

Answer 10

Every $2\times2$ real matrix is similar to a real matrix in one of these three forms: $$\pmatrix{a&0\cr0&b\cr},\qquad\pmatrix{a&1\cr0&a\cr},\qquad\pmatrix{a&b\cr-b&a\cr}$$

Answer 11

Let me point out that Vladimir Arnold was quite interested in similar question. He called the subject "mathematical trinities", see e.g. his paper "Symplectization, Complexification and Mathematical Trinities". As far as I remember from his lectures, his ideas were that many of these "trinities" are actually related to each other; and he also considered the subject as a tool to invent theories: see question marks at already cited "Arnold's table": jpg.

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Let me also mention some "trinities" which occur in my own research related to Capelli identities (which are some non-commutative analogs of $\det(AB)=\det(A)\det(B)$ ).

Matrix trinity - a) generic b) symmetric c) antisymmetric

Here how it goes in Capelli (and related Cayley) identities:

a) generic matrices - original Capelli identity has been discovered by Capelli in 19-th century - it is for "generic matrices" $A=x_{ij}$ $B = \partial_{ji}$

b) symmetric matrices - analog of the Capelli identity has been discovered by Turnbull around 1940-s - here $A=(x_{ij}+x_{ji})$ $B= \partial_{ji} + \partial_{ij} $.

c) antisymmetric matrices - analog has been found by Howe–Umeda and Kostant–Sahi around 1990, here $A=(x_{ij}-x_{ji})$ $B= \partial_{ji} - \partial_{ij} $.

Similar generalization were found for Cayley identity respectively: a) attributed to Cayley b) Gårding 1948 c) Shimura 1984 - see arXiv:1105.6270 for quite a complete information.

My question: is it really trinity ? Or you can propose some analogs of Cayley–Capelli for some other matrices, say "symplectic" ?

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It is might be strange, but other trinities like R,C,H also appears in the Capelli story - and they give different identities. Moreover trinities can be combined and we might get trinity^trinity^trinity...

Actually H-analog of the Capelli identity is not fully known for the momemnt - only analog for 1×1 matrices has been discovered quite recently by student of R. Borcherds, An Huang. Looking at this example I proposed some C-analogs of Capelli identities. Actually all generic/symmetric/antisymmetric can be complexified, hopefully there should exist quaternionic analogs and thus we might have trinity^trinity. Some partial results of trinity^trinity spirit for Cayley identity contained in loc. cit.

There are certain analogs of Capelli identities for classical Lie algebras: this can seen as gl/so/su trinity, well probably it is not the trinity in some strict sense. I have no idea can we have something like trinity^trinity^trinity ….

Answer 12

Previous MO questions:

• rational/trigonometric/elliptic trichotomy Groups, quantum groups and (fill in the blank)

• trichotomy of interrelated model structures: h-model, q-model, m-model Is the category of metric spaces and continuous maps Quillen equivalent to Top?

• "such "log-exp functions" are either eventually positive, eventually zero, or eventually negative. … It guarantees that the germs at infinity of such functions do indeed form a field K." Examples of sequences whose asymptotics can't be described by elementary functions

• a function of a complex variable with an algebraic addition theorem must be: 1) A rational function, 2) A rational function of e^{px}, or 3) A rational function of the Weierstrass elliptic function and its derivative. Trig functions based on convex curves

• "Every finitely generated infinite profinite group has a just infinite quotient. There is a trichotomy due to Wilson (and refined by Grigorchuk) describing what they can look like." What is the virtue of profinite groups as mathematical objects?

• "there is a trichotomy of curves given by g=0, g=1, and g≥2. If you look at topological, geometric, arithmetic properties of these curves, their properties align very strongly with these classes." Why should I believe the Mordell Conjecture?

• Kodaira dimension. κ(Y)<0, κ(Y)=0, κ(Y)=dimY. How "frequent" are smooth projective varieties with (anti-)ample canonical bundle?

• "Rank and period of primes in the Fibonacci sequence; a trichotomy," Fib. Quart., 45 (No. 1, 2007), 56-63). Can the difference of two distinct Fibonacci numbers be a square infinitely often?

• According to Etingof, Igor Frenkel once suggested that there are three "levels" to Lie theory, which could be called no loops, one loop, and two loops. Why can't we take three loops?

M.SE:

• "The set-theoretic setup of Categories for the working mathematician is somewhat subtle. … There is therefore a trichotomy of small sets, large sets, and proper classes. This is not the usual practice: we normally think of all sets as being small." Answer by Zhen Lin to confusion over the use of universes in category theory

• "There are three distinct aspects of schemes that each have their own purpose …." Answer by Michael Joyce to https://math.stackexchange.com/questions/99605/why-study-schemes/99615

TCS.SE:

• "… one of the most amazing facts about logic is that consistency strength boils down to the question "what is the fastest-growing function you can prove total in this logic?" As a result, the consistency of many classes of logics can be linearly ordered! If you have an ordinal notation capable of describing the fastest growing functions your two logics can show total, then you know by trichotomy that either one can prove the consistency of the other, or they are equiconsistent." Answer by Neel Krishnaswami to Axioms necessary for theoretical computer science

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A frequently cited paper: Caicedo and Ketchersid, "A trichotomy theorem in natural models of AD+", in "Set Theory and Its Applications", Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227–258.

Answer 13

An infinite finitely generated group $G$ has either 1, 2 or infinitely many ends; the last case is equivalent to saying that $G$ splits as an amalgamated free product or an HNN extension over a finite subgroup, by a theorem of Stallings. (Of course the assumption that $G$ is infinite brushes a fourth possibility under the carpet, namely that the number of ends is zero.)

Answer 14

Every isometry of a proper $CAT(-1)$ space is either elliptic, parabolic, or hyperbolic: elliptic means fixes a finite point; hyperbolic means fixes two infinite points connected by a translation axis, equivalently translation distance bounded away from zero; parabolic means translation distance limiting to zero but no fixed point. There are versions for proper Gromov hyperbolic spaces, and even for the nonproper case, if you are willing to "quasify" the statements of the cases, and if you are willing to let the trichotomy degenerate to a dichotomy.

Every isometry of Teichmuller space is either elliptic, parabolic, or hyperbolic. This is Bers' form of Thurston's trichotomy for mapping classes: finite order, reducible, pseudo-Anosov. This trichotomy also has an interpretation in terms of the action of the mapping class group on the curve complex which is a nonproper Gromov hyperbolic space by a theorem of Masur and Minsky.

For elements of $Out(F_n)$, the outer automorphism group of a rank $n$ free group, there are related trichotomies and other -otomies coming from the work of Bestvina, Feighn, and Handel on relative train track theory. The simplest one is that every element of $Out(F_n)$ is either of finite order, or of polynomial growth, or of exponential growth.

Answer 15

Knots are either torus, satellite or hyperbolic.

Answer 16

Every finitely generated group is either of polynomial growth, intermediate growth, or exponential growth.

As a statement, there is not much to this, the only mathematical content is that the growth function of every finitely generated group has an exponential upper bound.

But as a method of classifying finitely generated groups, it has been very fruitful: Gromov's theorem on groups of polynomial growth; the incredibly rich theory that arose from Grigorchuk's original construction of an intermediate growth group; and the emergence of rich classes of exponential growth groups such as word hyperbolic groups.

Answer 17

The reduction (special fiber) $E_{\mathfrak p}$ of (the Neron model of) an elliptic curve $E$ modulo a prime ${\mathfrak p}$ is one of:

1. good reduction = stable reduction = $E_{\mathfrak p}$ is non-singular
2. multiplicative reduction = semi-stable reduction = $E_{\mathfrak p}$ is a product of the multiplicative group times a finite group
3. additive reduction = unstable reduction = $E_{\mathfrak p}$ is a product of the additive group times a finite group

Of course, this trichotomy is a reflection of the fact that there are only three sorts of connected one-dimensional Lie groups, namely the additive group, the multiplicative group, and the compact case (elliptic curves).

Answer 18

The Gelfand-Kirillov dimension of an algebra is either zero, one, or at least $2$.

This sounds silly, but one has to remember that the GKdim is a real number (or $\infty$) and that there are algebras with GKdim equal to $r$ for all real $r\geq2$.

This is a combination of results of several people. See the books by McConnell and Robson, or by Krause and Lenagan.

Answer 19

The endomorphism ring of an elliptic curve is either $\mathbb{Z}$, an order in a quadratic field, or an order in a quaternion algebra (ranks $1,2$, and $4$, respectively).

Answer 20

In thermodynamic formalism for dynamical systems, a Hölder continuous potential function $\phi$ on a countable state topological Markov chain $(X,\sigma)$ is either positive recurrent, null recurrent, or transient. These correspond to the three possibilities for equilibrium states (shift-invariant measures maximising the quantity $h(\mu) + \int_X \phi\,d\mu$): existence of a finite equilibrium state is equivalent to positive recurrence; null recurrence is the boundary case where the equilibrium state becomes $\sigma$-finite but not finite, and transience is the case where there is no equilibrium state (all the weight has gone to infinity).

These can be characterised in terms of a particular sequence $a_n>0$: positive recurrence is equivalent to $\limsup a_n > 0$, null recurrence is equivalent to $a_n\to 0$ and $\sum a_n=\infty$, and transience is equivalent to $\sum a_n<\infty$. I imagine this trichotomy for sequences appears in other places as well.

Edit: It's worth mentioning that this trichotomy is also true for random walks on directed graphs (weighted or unweighted) -- historically I believe this is where it was first studied and where the terminology came from, but as a dynamicist I more immediately think of the interpretation above. In this setting the interpretations are as follows:

• Positive recurrent -- with probability 1, a random walk returns to where it started, and the expected return time is finite.
• Null recurrent -- the walk returns to the starting position with probability 1, but the expected return time is infinite.
• Transient -- with probability 1, the walk never returns to its starting position.

Answer 21

Every prime of a field either ramifies, splits or is inert in a field extension.

Answer 22

Every logical statement is either true, undecidable or false.

Answer 23

A local domain can be pure characteristic $0$, pure positive characteristic, or mixed characteristic. (Meaning that the algebra has characteristic $0$ but its residue field has positive characteristic.)

Of course, this being a trichotomy relies on the fact that it's a domain; otherwise you could also have characteristic $p^n$ (with the residue field having characteristic $p$). I don't really know how that case is classified (I guess it's a form of mixed characteristic)...

Answer 24

Let $G$ be a complex reductive algebraic group and $F$ a finite rank free group. In the moduli space of $G$-representations of $F$, namely the character variety $\operatorname{Hom}(F,G)//G$, the regularity of points are (generally, but not always) classified as follows:

1. Smooth points: good representations (irreducible with central stabilizer)

2. Orbifold singular points: bad representations (irreducible with non-central stabilizer)

3. non-Orbifold singular points: reducible representations.

See here, here, and here for more information.

Answer 25

A simply laced Dynkin diagram is A, D, or E. (This is perhaps implicit in Alexander Chervov's answer but I think deserves to be explicitly stated.)

Answer 26

Here is a cluster of examples with a common theme, based partly on comments here and in an MO thread on whether an empty space should be considered connected, and partly on an article in the nLab, "too simple to be simple".

One should note that some of these trichotomies were once-upon-a-time considered dichotomies; for example, for many people in the past, 1 was a prime number. Also one should notice that there are a number of cross-connections (homomorphisms, if you will) between these examples, and that list is by no means complete (this is CW, so feel free to add more!).

• A module can be reducible, irreducible, or zero.

• A filter in a Boolean algebra can be "submaximal" (faute de mieux!), a maximal filter = ultrafilter, or an improper filter.

• An element in a p.i.d. is composite, prime, or a unit.

• A topological space (or a graph) can be disconnected, connected, or "unconnected" (empty).

• As for elements satisfying a predicate, one can have a multiplicity, unique existence, or nonexistence.

The last trichotomy is often considered a dichotomy: nonuniqueness vs. uniqueness. (I.e., nonexistence falls under the scope of uniqueness = "at most one".) But experience in mathematics, e.g. in category theory and its focus on universal properties, shows that unique-existence deserves to be considered in a category of its own.

• Theories can be incomplete but consistent, complete, or inconsistent.

Answer 27

Did anybody mention yet the different arithmetic-wise behaviour of algebraic curves of genus $0$, $1$, or $\geq2$?

Answer 28

In number theory, there is the trichotomy $\infty$, odd $p$, and $2$ (the oddest prime).

Answer 29

Suppose that a group $G$ has an action on a tree with no inversions and no global fixed point. Then either (a) $G$ is expressible as a (non-trivial) amalgamated free product; (b) $G$ is indicable (i.e. it maps onto an infinite cyclic group); or (c) $G$ is expressible as the union of a strictly ascending sequence of subgroups.

The negation of this property (every action of $G$ on a tree without inversions has a fixed point) is called FA by (J.-P.) Serre.

A more mundane, but related, example: a single automorphism of a tree either fixes a point, inverts an edge, or has an invariant (doubly-infinite) line contained in the tree on which the automorphism acts by translations.

Answer 30

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.)