
A Proof About Where Symmetries Can’t Exist - jonbaer
https://www.quantamagazine.org/a-proof-about-where-symmetries-cant-exist-20181023/
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Sniffnoy
If you know a bit of math, you'll probably get a better understanding by just
reading the first page of the paper[1] rather than reading this article.

[1] [https://arxiv.org/abs/1710.02735](https://arxiv.org/abs/1710.02735)

~~~
Sharlin
I definitely think I know "a bit of math" (including group theory and
topology) and the first page was beyond my grasp, although not by a large
margin.

~~~
gjm11
[Comment number 2, describing what this theorem says for people with patience
but not necessarily much mathematical expertise. Read comment number 1 first.
There isn't a number 3.]

OK, time for _ingredient number 2_ which is the idea of a _manifold_. Roughly,
an n-dimensional manifold is a geometrical object which, whenever you look at
it closely enough, looks like "ordinary" n-dimensional "Euclidean" space. Once
again, let's begin in dimension 2. "Ordinary" 2-dimensional space is something
mathematicians write as R^2: jus as Z^2 consisted of _pairs of integers_ , R^2
consists of _pairs of real numbers_. (Real numbers are what you probably think
of just as "numbers".) We can think of this as an infinite plane; the two
numbers are coordinates. Now, for a fairly typical example of a 2-dimensional
manifold, consider a sphere -- the surface of the earth, say, if you smooth
that out a bit. Any small portion of it looks just like a small portion of the
plane, which is why there is a Flat Earth Society. But the whole thing has a
different structure -- e.g., the sphere is finite in extent in a way the plane
isn't. Another example is the surface of a ring do(ugh)nut. This is also
finite in extent, but it turns out to be genuinely different from the sphere,
and there's a whole lot of interesting mathematics around that, which I am
going to ignore here.

Now, just as we considered those very special transformations acting on Z^2
before, we are going to consider transformations acting on a manifold: things
that pair up each point on the manifold "before" the transformation with some
point "after" it. For instance, suppose we take the (idealized) surface of the
earth; one example of a transformation would be what you get by rotating the
earth about its axis through 10 degrees so that every point moves west (and
e.g. Kansas City lands more or less on top of Denver). But our transformations
can be more complicated: imagine our sphere to be a thin sheet of very
flexible rubber forced somehow to remain on the surface of a globe; we can
then push things around however we like provided the sheet never tears or
overlaps itself. These things (with some technical restrictions I won't go
into) are called _diffeomorphisms_. And, though it's harder to visualize, we
can do much the same thing with any manifold of any dimension. If M is our
manifold then we call the set of all its diffeomorphisms Diff(M). And, just
like SL(n,Z), this thing is a _group_ : we can compose two diffeomorphisms to
get another diffeomorphisms, and because we insisted on no tearing or double-
covering every diffeomorphism can be undone by another diffeomorphism.

All right, nearly there. _Ingredient number 3_ is the idea of a _group
homomorphism_. Suppose we have two groups; call them G and H. Suppose that for
each element of G we somehow pick out an element of H. We'll write f(g)=h to
mean that element g in G yields element h in G; "f" is the name we're giving
to our correspondence between G and H. Unlike the transformations considered
above, we aren't going to insist on any sort of invertibility; f might map
lots of different g to the same h, and there might be some h that aren't the
"image" of any g. Here's a simple example: consider SL(2,Z) again, and
consider a manifold M that's just ordinary 2-dimensional space, what we called
R^2. Everything in SL(2,Z), remember, maps (x,y) to (ax+by,cx+dy) -- and we
can do _that_ just as well when x,y are arbitrary real numbers as when they
are integers, and the resulting thing is in fact a diffeomorphism. So
everything in SL(2,Z) gives rise to a thing in Diff(R^2).

In this case, because these are "the same" transformation in some sense,
composition of things in SL(2,Z) matches up nicely with composition of things
in Diff(R^2). This sort of matching-up-nicely can happen even in less
straightforward cases. If f(g1 compose g2) always equals f(g1) compose f(g2)
then we call f a _group homomorphism_ between SL(n,Z) and Diff(M). The
specific transformations in SL(n,Z) and in Diff(M) needn't have anything much
to do with one another -- but the _relationships between them_ need to have
compatible structures, in some sense.

So, we saw above that you can embed a copy of SL(2,Z) inside Diff(R^2). More
generally, there's a copy of SL(n,Z) inside Diff(R^n). What these guys have
done is to _put some limits on correspondences of this kind between SL(n,Z)
and Diff(R^m) where m is smaller than n_ : the idea is that SL(n,Z) is an
n-dimensional thing and that you can't squash it into something of much lower
dimension without destroying its structure.

Unfortunately there's one more bit of technical detail needed before stating
their result. _Ingredient number 4_ is the idea of a _finite-index subgroup_.
Zimmer's conjecture restricts not only group homomorphisms from SL(n,Z)
itself, but also from certain smaller things that in some sense contain most
of the structure of SL(n,Z). So, suppose you have some subset S of SL(n,Z)
which is also a group: composites and inverses of things in S are always in S
themselves. Then it turns out that SL(n,Z) can be partitioned into "copies" of
S. One is S itself; if x is anything that isn't in S, then the compositions "x
compose s", as s runs over everything in S, form another of these copies, and
x itself is in that copy. Any two of these copies are either identical or
disjoint, it turns out; so the whole of SL(n,Z) is made up of a bunch of these
things. If there are only _finitely many_ of these disjoint copies, we say
that S has "finite index"; so e.g., maybe there are 12 of them, meaning that S
is in some sense 1/12 as big as all of SL(2,Z). (Just to be clear, SL(n,Z) is
_infinite_.)

We can finally state the theorem properly: If S is a finite-index subgroup of
SL(n,Z), and f : S -> Diff(M) is a group homomorphism, and the dimension of M
is m where m < n-1, _then_ the image of f -- the set of things in Diff(M)
corresponding to things in S -- is _finite_.

That is: you can't fit something with the same structure as "most of" SL(n,Z)
into the diffeomorphisms of something with dimension < n-1, unless you
collapse that structure "almost completely".

~~~
Sniffnoy
Sorry, just a posting note: If you want your comments to display in the right
order, it would make more sense to have your second comment be a reply to your
first, rather than having them as sibling comments.

