The boundary a growing region forgets

“Six point oh two atoms per cubic nanometre” is an average over a region, and the number is quoted as though the region were not part of it. Take a bigger box, take a sphere instead, take a slab — the answer is supposed to be the same. It usually is, and the reason has nothing to do with atoms.

How fast a group grows counts the elements of a plane group spellable in at most R letters and finds R²: the group reporting the dimension of a plane it has forgotten. This essay takes one more difference, and the answer decides whether averages exist.

The question has a shape worth naming before any counting. Two large regions that share most of their volume must give nearly the same average of any bounded quantity, since the parts they do not share are small. So the whole issue is whether a sequence of regions can be made to share more and more of itself as it grows — which is a question about how much of a region is near its edge, and about nothing else.

The boundary is one dimension down #

The sphere of radius R in a group is the set of elements at exactly that distance, and it is the difference of two consecutive balls. If balls grow like R² then spheres grow like R, so the fraction of a ball that lies on its own boundary falls like 1/R.

A sequence of regions whose boundary share goes to zero is a Følner sequence, and a group with one is called amenable. The name is unfortunate and the property is exactly what an average needs: if the boundary is negligible then two large regions that overlap in most of their volume give nearly the same average, and shifting a region by one step changes the answer by an amount proportional to the boundary — which is to say, by nothing in the limit.

That is the theorem behind every intensive quantity a crystal has. A density, a composition, an average scattering power, the mean of anything bounded and periodic: each is an average over a region, and each has a value independent of the region because the group of translations is amenable.

It is worth being precise about what “negligible” has to mean, because the naive version is not enough. It is not that the boundary is small compared with the region — a slab’s boundary is a fixed fraction of it, which is small in the sense of being less than a half and useless in the sense that matters. The condition is that the fraction goes to zero, so that any fixed amount of error attached to the boundary is eventually beaten by any fixed amount of signal in the interior. A fraction that settles at two sevenths or at two thirds never gets beaten.

The other thing worth saying is why the boundary is where the trouble lives. Take a region and shift it by one generator. Everything that was interior stays inside; only the boundary can enter or leave. So the change in an average under a shift is bounded by the boundary’s share times the range of the function, and a function that is bounded and a boundary that vanishes make the average shift-invariant in the limit. Shift-invariance is what “belongs to the crystal rather than to the window” means, and it is the whole content.

The seven groups in that table were chosen to span the seventeen rather than to make a point: p1 has no point symmetry at all, p6m has the most, and the rest sit between. Their constants differ by about ten per cent and their behaviour does not differ at all, which is the answer to the obvious worry — that a group with more operations might somehow have a fatter boundary. It does not, because the extra operations multiply the ball and the sphere by the same bounded factor.

Where it fails, it fails completely #

The property is not automatic and the failure is not marginal.

There is one more consequence of the polynomial growth that is worth extracting because it is peculiar to lattices. If balls grow like R^d then the ratio of consecutive balls tends to one, so a ball and the ball one step larger are nearly the same region. In an exponentially growing group they are not: the larger ball is three times the smaller one, and no two consecutive regions in that sequence resemble each other at all. Amenability is that resemblance, and polynomial growth is the cheapest way to have it.

There is a second reason to run the free group rather than merely to cite it. The two curves are the same measurement — outermost shell over everything inside — computed by the same code on two Cayley graphs, so the difference between them cannot be an artefact of how the counting was set up. A comparison between a measured quantity and a quoted one always leaves the possibility that the two were defined differently; a comparison between two measurements of the same quantity does not.

The contrast is worth holding onto because the two groups are not exotic relatives. The free group on two generators is the group of two symbols and their inverses with no relations at all — the simplest possible presentation after the trivial one — and it is the group a group in four letters starts from before any relation is imposed. Adding relations is what makes a group small enough to have averages, and a plane group is the free group with enough relations imposed to make it grow polynomially.

There is a temptation to dismiss the free group as a curiosity with no crystallographic relevance, and it should be resisted for one reason: it is the group of a tree, and a tree is what a Cayley graph looks like when there are no relations to close a loop with. Every relation a group has is a cycle in its Cayley graph, and cycles are what stop the graph branching away from itself. So the difference between the two curves above is the difference between a graph that folds back on itself and one that does not, and a crystal is the extreme case of folding back: its Cayley graph is a lattice with a bounded decoration, and it embeds in the plane.

That reading also says which groups will be on which side. A group acting on a space of non-positive curvature with genuine branching — a free group, a surface group of high genus, the group of a hyperbolic tiling — grows exponentially and keeps its boundary. A group that is a lattice in a Euclidean space grows polynomially and loses it. Past two, the list does not stop is where this collection meets the hyperbolic case, and the boundary behaviour is the analytic shadow of the curvature there.

One more property of the seventeen deserves a sentence, since the table shows it without saying it. The seven groups differ in their constants and not in their exponent: the ratio times the radius settles at between three and four depending on the group, and the exponent is one in every case. That is the group’s own symmetry showing up as a constant while the dimension shows up as the power — the same division how fast a group grows finds one level down, where the leading coefficient of the quadratic depends on the group and the degree does not.

Growing is not enough #

Between the two extremes there is a case that matters much more for crystallography, and it is available inside the ordinary lattice.

The translation group of a crystal is amenable, so Følner sequences exist in it — but that does not mean every growing sequence of regions is one. A slab seven layers deep and a hundred wide has a hundred thousand points and two faces of ten thousand each, and widening it does not change that proportion at all.

That table is the point of the essay. The failure is not noise and it does not shrink; it is a definite number, it is the same number at every width, and which average a region gets wrong is decided by the direction in which the region fails to grow.

The arithmetic behind the seventh is worth doing once, because it generalises. A slab of half-thickness t has 2t + 1 layers, of which t + 1 are even and t are odd when the centre is even — so the average of the alternating function is 1/(2t + 1) with the sign of whichever parity is in the majority. For t = 3 that is 1/7. Making the slab thicker shrinks the error like one over the thickness, which is the right behaviour and is a different limit from making it wider: widening does nothing at all and thickening is what converges. A measurement improved by taking a larger area when the error is set by the depth is a measurement that will not improve.

The needle row is worth reading as a second case rather than as a repetition. A needle three units across and long in k gets the layered function right — it is long in the direction that function varies in — and gets a function varying along i wrong by the same seventh. So neither region is simply “bad”; each is exact for some quantities and definitely wrong for others, and the split is decided by comparing the direction the function varies in with the direction the region is thin in. That is a more useful statement than “small samples are unreliable”, because it says which measurements on a given sample are safe.

What this says about measuring a crystal #

Three things follow, and the third is the one worth carrying.

The first is that a thin film’s composition is a different quantity from a bulk composition, not a noisier estimate of it. A measurement averaged over a region that is not Følner returns a number that belongs to the region, and no amount of widening the film converges it onto the bulk value. That is a statement about geometry and it holds before any question of surface reconstruction or contamination.

The second is that the direction matters. A slab gets a layered property wrong and an in-plane property right; a needle does the reverse. So a measurement can be simultaneously reliable for one quantity and systematically wrong for another, on the same sample, and the two failures are not correlated with each other in any way a statistical treatment would find.

The third is the connection to a sum. A conditionally convergent lattice sum has no value until the summation order is named, and the cube and the sphere disagree — in the same group, over regions that are both Følner. So amenability is necessary and it is not sufficient: it makes averages of bounded quantities shape-independent, and says nothing about a sum whose terms are not summable. The two essays are about the two separate ways a shape can get into an answer, and a reader who has met only one of them will attribute the wrong failure to the wrong cause.

A last framing, since the word amenable has appeared several times without doing much work. The property has a dozen equivalent definitions — an invariant mean, a Følner sequence, no paradoxical decomposition, a fixed point for every affine action — and this essay uses one of them, the one that is a count. The others are theorems relating it to the count, and none of them is proved here. What is worth knowing is that the count is not a proxy for the property; it is one of its definitions, so measuring it settles the question rather than providing evidence about it.

The last of those is the one that keeps the free group’s number honest. 4·3^{R−1} is a formula anyone can write down and get slightly wrong — an off-by-one in the exponent gives a plausible sequence with the wrong limit — so the count is done twice, once by the formula and once by walking the tree of reduced words, and the two are required to agree at every radius. A closed form that has never been checked against an enumeration is a conjecture with confident typography.

One reading of the whole essay is worth setting down because it changes what several other results mean. This collection contains a great many statements of the form “the average over the crystal is such-and-such” — the symmetry of an average, the average that makes it finite, Neumann’s principle in every one of its appearances. Each of those is an average over a growing region, and each is silently using the fact that the region does not matter. The fact is true, it is a property of the translation group rather than of the physics, and it is measured here rather than assumed. That is the point of doing it: a step used everywhere and stated nowhere is the kind of step that is wrong in the one case nobody checked.

There is one apparent paradox in the table which is worth resolving rather than leaving to trouble a reader. A cube of half-width R gets the layered average wrong by 1/(2R+1) — a slab’s error, with the thickness equal to the width — and that is not a failure of the cube: the error goes to zero because the cube’s thickness grows along with everything else. The slab’s error is the same expression with the thickness frozen. So the two cases are one formula with one parameter held fixed or not, and “Følner” is the statement that every direction’s extent goes to infinity, rather than that the volume does.

That also explains why the ball does better than the cube at the same radius. A ball has no flat face to hold an imbalance, so its layer counts are nearly balanced by the curvature of its boundary rather than only by its extent, and the residual is smaller by an order. Both go to zero; the constant is a property of the shape.

One more consequence, and it is the one that makes the whole thing a crystallographic statement rather than a group-theoretic one. Every space group contains a lattice of finite index, and a subgroup of finite index in an amenable group is amenable — so every one of the two hundred and thirty is amenable, without any of them being checked. The property is inherited from the translations, and the point operations, which are finitely many, cannot disturb it. That is why a density is a well-defined quantity for a crystal of any symmetry, and why nothing in the classification of space groups ever has to worry about it.

Where this stops #

Two limits, both about scope rather than about correctness.

The measurement here is of ball and sphere sizes in a Cayley graph, which is a statement about the group with a chosen generating set. A different generating set gives different balls and a different constant in the 1/R, and the property of being amenable does not change — that independence is a theorem and this collection does not prove it. What is computed is the behaviour for the generators the presentations use, and the reader should read the constants as belonging to that choice.

And amenability is decided here for two groups by exhibiting the behaviour, which settles those two cases and no others. There is no procedure in general: whether a given finitely presented group is amenable is undecidable, which is the same shape of result as no procedure decides a tiling and comes from the same source. The groups a crystal supplies are all amenable for the same reason — they contain a lattice of finite index and grow polynomially — and that reason is available here as a measurement rather than as a proof.