Straight lines, and no distances

Every finiteness met so far comes back to one trick. The average that makes it finite is the whole of it: sum a quadratic form over a finite group and the result is preserved by every member, so a finite group of integer matrices has a metric it is orthogonal in, and a finite group of motions has a point it fixes. From there the classification is a classification of subgroups of the orthogonal group, and the bound that makes the list finite is a statement about matrices preserving something.

What forces a lattice leans on the same thing one level up. Its central inequality — the commutator of two rotations is much smaller than either — is stated in an operator norm, and its own refusal table says what happens without one: a stretch that doubles one axis and halves the other, and a shear, each sit at distance one from the identity, and their commutator sits at distance three against a permitted two. “The metric is part of the statement, not a detail of the proof.”

So the question is what survives when the motions are replaced by affine maps — any invertible linear part, any translation, no orthogonality asked. That is the largest group of transformations still carrying straight lines to straight lines and lattices to lattices, which is everything a crystal’s description uses except the distances.

Bieberbach’s first theorem does not survive. It is false in the plane, and the counterexample is two lines long.

Two maps, and an orbit on a parabola #

Take A(x, y) = (x + 1, y + x + ½) and B(x, y) = (x, y + 1), and check they commute, which they do exactly: both compositions send a point to (x + 1, y + x + 3/2). So the group they generate is abelian, generated by two elements of infinite order with no relation between them — abstractly the plane’s own group of translations, ℤ².

Its elements are A^n B^m, whose linear part is the shear by n and whose translation is (n, m + n²/2). The image of the origin is therefore the point (n, n²/2 + m), and the set of all of them is the whole-number grid sheared onto a parabola.

That orbit is discrete. Two images with different first coordinates are at least one apart in x; two with the same first coordinate are at least one apart in y. The closest any two distinct images come is exactly one, which is measured on nine hundred and forty-nine of them rather than argued.

And it leaves no point far away. Every point of the plane has an image of the origin within about six-tenths of it — measured on a grid of sixteen hundred probes over the unit square, with the worst at 0.610 — so the quotient is compact.

Those are the two hypotheses of Bieberbach’s first theorem, both satisfied. What the theorem concludes is that the group contains two independent translations of finite index. It does not.

An element of the group is a translation exactly when its linear part is the identity, which means n = 0. So the translations are the powers of B and they span a single direction. Rank one, where the conclusion requires rank two, with both hypotheses met and the only difference being that the elements are not motions.

No change of coordinates repairs it #

The obvious objection is that this must be a group of motions in disguise — that some clever coordinates turn the shears into rotations. It is not, and the argument is short enough to be worth making rather than announcing.

An affine change of coordinates conjugates the whole group, and conjugation by an affine map carries translations to translations: if g has identity linear part then so does hgh⁻¹, since linear parts multiply. So the rank of the translation subgroup is unchanged by any affine coordinate change whatever. Any action of ℤ² on the plane by motions, discrete with compact quotient, has translations of rank two by Bieberbach’s theorem. One is one and two is two.

The same conclusion arrives from the other direction, and this is the version that says what has actually broken.

A form Q is preserved by a linear map L when LᵀQL = Q, which is three linear equations in the three entries of a symmetric two-by-two form. Stacking those equations for the first several shears and eliminating leaves a solution space of dimension one, spanned by the form with a single one in its top corner. That form measures the horizontal coordinate and calls every vertical displacement zero — determinant nought, positive semi-definite and degenerate.

There is no metric for the group to preserve. Not an unusual one, not a badly conditioned one: none.

The average has nothing to converge to #

That is not merely the absence of a preserved metric; it is the failure of the construction that produces one.

Sum MᵀM over a finite group, divide by the order, and the result is preserved by every member — one line of algebra, and it is exact here on a group of three integer matrices of order three, to the last digit the arithmetic carries. The same sum over the shears [[1,0],[n,1]] for n from −K to K does not settle: its leading entry is 1.7 at K = 1 and 25.0 at K = 8, growing as K² because the shears themselves grow.

An average over an infinite group of unbounded matrices is not a large finite average. It is a divergent sum, and there is nothing at the end of it to call a metric. The degenerate solution space and the divergence are the same fact seen twice: the form the group preserves is the limit of those averages after rescaling, and rescaling a growing sum to make it converge is what flattens it onto the horizontal axis.

The inequality is not nearly true #

With no metric, the inequality the whole proof turns on has no statement.

For orthogonal matrices in any dimension the ratio never passes two, and for rotations of space it never passes one, which is what makes nested commutators shrink and is the lever the general proof pulls. Sampled over invertible matrices with no orthogonality asked, 552 of 4,000 pairs pass two and the largest found is 8.76. The named pair — the stretch and the shear from the refusal table below — comes out at exactly three.

So the statement is not weakened by dropping the metric; it is destroyed. Nested commutators against an affine map need not shrink, there is no neighbourhood of the identity in which they must eventually vanish, and the argument that produces translations from small elements has no small elements to work with.

One dimension up, the same construction has a name #

The plane’s example is the smallest of a family, and the next member is a group already turned away once here.

Take the discrete Heisenberg group — triples of whole numbers with the product (a, b, c)(a′, b′, c′) = (a + a′, b + b′, c + c′ + a b′) — and let it act on three-space by the affine maps its matrix form already is: a unipotent linear part and a translation. The action is discrete, its quotient is compact, and the quotient is a closed three-manifold carrying a flat affine structure that no flat metric produces. The clauses a crystallographic group must satisfy turn this group away for having a maximal abelian normal subgroup of infinite index, and the action is why the near-miss is worth having rather than being a curiosity of presentation.

The plane’s example is the abelian case of the same thing. There the group is ℤ² rather than the Heisenberg group, and the failure shows up in the rank of the translations rather than in the index of a subgroup; the construction — an abelian or nilpotent group given an action by shears rather than by slides — is identical. Going up one more dimension gives a longer list of such structures, and classifying them is a subject of its own rather than a corollary.

What none of them is, is a crystal. A structure on which the Heisenberg group acts this way has no unit cell in the ordinary sense: the operation that carries one repeat to the next is not a slide, so the second repeat is sheared against the first and the tenth is sheared by ten times as much. Nothing periodic looks like that, which is the physical statement of the rank-one failure measured above.

What survives, and what nobody has settled #

Something does survive, and it is much weaker than a lattice.

The example above is abelian, so it is about as tame as a group can be, and it is consistent with everything known. Auslander’s conjecture of 1964 says that a group acting on affine n-space properly discontinuously and with compact quotient must be virtually solvable — that is, must have a solvable subgroup of finite index. That is the affine replacement for the lattice, and it is much less than a lattice: a virtually solvable group can have translations of any rank from nought upwards, as the example shows.

The conjecture is proved in dimensions one and two by elementary means, in three by Fried and Goldman in 1983, and up to six by Abels, Margulis and Soifer. Above six dimensions it is open, and has been for sixty years.

The related question that was settled, and settled the other way, is Milnor’s. He asked in 1977 whether a group acting properly on affine space — dropping the compactness of the quotient — must be virtually solvable, and Margulis answered no in 1983 by constructing a free group of rank two acting properly on three-dimensional affine space. There is no analogue of that among groups of motions: a discrete group of motions of three-space acting properly is virtually solvable or is not discrete, because the averaging argument and the commutator inequality between them forbid a free group. Margulis’s example is what the metric was preventing, and the quotients it produces are now their own subject.

The classification was already affine, which is the confusing part #

There is a genuine tension here with something said elsewhere in these pages, and leaving it unremarked would be worse than stating it.

The classification of crystallographic groups is a classification up to affine equivalence. Two space groups are the same group when some affine map carries one onto the other, not only when some motion does — that is why there are two hundred and thirty and not more, and why the eleven enantiomorphic pairs need the affine maps to be restricted to the orientation-preserving ones before the count moves to two hundred and nineteen. A cell is not a shape in that classification; it is a basis, and which lattice type a group has is a statement about integer matrices rather than about angles.

So affine maps are already the right notion of sameness, and a reader may fairly ask how the metric can be doing work if the answer does not depend on it.

The resolution is that the two uses of “affine” are at different places in the argument. The groups being classified are groups of motions, and the equivalence used to compare them is affine. That is a quotient taken at the end. What this page does is different: it lets the group itself consist of affine maps, which changes what the objects are rather than which of them count as the same. A group of motions has a preserved metric whatever coordinates it is written in, because a conjugate of an orthogonal group is orthogonal in the conjugated metric; a group of affine maps need not have one in any coordinates at all, and the example above does not.

The lattice is affine data and the point group is not. A lattice is a discrete subgroup of translations and no distance enters its definition; the fourteen types are separated by which integer matrices preserve them, which is arithmetic. A point group, by contrast, is a finite group of linear maps, and finiteness is what averaging converts into orthogonality. Take the finiteness away — which is exactly what an infinite group of shears does — and the linear parts stop being a point group in any sense, and there is nothing left for the metric to be recovered from.

That is the sharpest statement of what the metric supplies. It does not supply the lattice, the cell or the classification’s equivalence. It supplies the finiteness of the linear parts, and everything else follows from that.

Where the exactness stops #

Computed here. That the two generators commute, on five points, exactly. The orbit of the origin out to reach six — 949 images — with the closest distance between two of them and the farthest any point of the unit square lies from one, over sixteen hundred probes. Which elements are translations. The solution space of forms preserved by the first four shears, by elimination. The averaged form over a finite group and its residues, and the same average over the shears to eight terms. And the commutator ratio on four thousand random pairs, with the named pair computed exactly.

The measurements are of finite pieces. Discreteness is checked on 949 images and compactness on the unit square; both statements are about the whole plane and both follow from the closed form of the orbit, which is what the measurement is checking rather than replacing.

Every row of the status table is quoted. Auslander’s conjecture, its proofs up to six dimensions and Margulis’s example are results in the literature. Nothing on this page proves or reproduces any of them, and the only thing derived here is the plane’s counterexample to the conclusion of Bieberbach’s first theorem.

And the counterexample is to the conclusion, not to the theorem. Bieberbach’s hypotheses include that the elements are isometries. This group’s elements are not, so no theorem is contradicted; what is shown is that the hypothesis is doing work rather than being a convenience. That distinction is the whole of the essay and is easy to lose.

The first refusal is the objection this essay exists to answer. The second is the inequality applied where it has no statement, and it is the more useful of the two, because a reader who has followed the averaging argument will have the bound in mind as though it were a fact about matrices rather than a fact about matrices preserving something.

Who asked, and who answered which half #

Louis Auslander stated his conjecture in 1964, in a paper on the fundamental groups of compact complete flat affine manifolds — which is what the quotients in question are. John Milnor asked the wider question in 1977, in a paper explicitly listing what was known and what was not, and conjectured the answer was yes; Gregory Margulis produced the free group in 1983 and the answer was no. Fried and Goldman settled three dimensions the same year by classifying the possible linear parts.

The example on this page is much older than any of that, and it is nobody’s in particular. A group of unipotent affine maps acting simply transitively on the plane is the smallest affine structure on a nilpotent group, and such structures were being written down from the 1950s onwards, usually as examples in a classification rather than as counterexamples to anything. What makes it worth the space here is the direction it is read in: the classifications wanted to know which manifolds carry such structures, and the question on this page is which hypothesis of a theorem about crystals is load-bearing.

Still open: how much of the restriction is metric #

The crystallographic restriction says a lattice admits rotations of order two, three, four and six and nothing else, and its proof is one line of integer arithmetic on a trace. Nothing in that line mentions a metric.

So the natural question the affine case raises is which of the results so far are about distance and which are about integers. The restriction is arithmetic: an integer matrix of finite order has a trace that is an integer and an eigenvalue that is a root of unity, and neither statement needs a norm. The finiteness of the list is metric, since it rests on averaging. Bieberbach’s first theorem is metric, as this page shows. Where the reduction modulo three sits is less clear: its argument is about integer matrices of finite order and mentions no metric anywhere, but the finiteness it proves is the finiteness of a group that averaging has already supplied.

Sorting the collection’s results into the two kinds is a piece of work nothing here has done, and the reason it would be worth doing is that the arithmetic half transfers to the affine case and the metric half does not. A crystallographic group in the affine sense still has integer linear parts if it preserves a lattice, and what the restriction then says about it is exactly what it says about a crystal — while everything about how many such groups there are becomes an open question again.