Tag: John Pardon

On Tao’s talk and the 3-dimensional Hilbert-Smith conjecture

Conan1 Comment

Last Wednesday Terry Tao briefly dropped by our little town and gave a colloquium. Surprisingly this is only the second time I hear him talking (the first one goes back to undergrad years in Toronto, he talked about arithmetic progressions of primes, unfortunately it came before I learned anything [such as those posts] about Szemeredi’s theorem). Thanks to the existence of blogs, feels like I knew him much better than that!

This time he talked about Hilbert’s 5th problem, Gromov’s polynomial growth theorem for discrete groups and their (Breuillard-Green-Tao) recently proved more general analogy of Gromov’s theorem for approximate groups. Since there’s no point for me to write 2nd-handed blog post while people can just read his own posts on this, I’ll just record a few points I personally found interesting (as a complete outsider) and moving on to state the more general Hilbert-Smith conjecture, very recently solved for 3-manifolds by John Pardon (who now graduated from Princeton and became a 1-st year grad student at Stanford, also appeared in this earlier post when he gave solution to Gromov’s knot distortion problem).

Warning: As many of you know I never take notes during talks, hence this is almost purely based on my vague recollection of a talk half a week ago, inaccuracy and mistakes are more than possible.

All topological groups in this post are locally compact.

Let’s get to math~ As we all know, a Lie group is a smooth manifold with a group structure where the multiplication and inversion are smooth self-diffeomorphisms. i.e. the object has:

1. a topological structure
2. a smooth structure
3. a group structure

It’s not too hard to observe that given a Lie group, if we ‘forget’ the smooth structure and just see it as a topological group which is a (topological) manifold, then we can uniquely re-construct the smooth structure from the group structure. From my understanding, this is mainly because given any element in the topological group we can find a unique homomorphism of the group \mathbb{R} into the manifold, sending 0 to identity and 1 to the element. resulting a class of curved through the identity, a.k.a the tangent space. Since the smooth structure is determined by the tangent space of the identity, all we need to know is how to ‘multiply’ two such parametrized curves.

The way to do that is to ‘zig-zag’:

Pick a small \varepsilon, take the image of \varepsilon under the two homomorphisms, alternatingly multiplying them to obtain a sequence of points in the topological group. As \varepsilon \rightarrow 0 the sequence becomes denser and converges to a curve.

The above shows that given a Lie group to start with, the smooth structure is uniquely determined by the topological group structure. Knowing this leads to the natural question:

Hilbert’s fifth problem: Is it true that any topological group which are (topological) manifolds admits a smooth structure compatible with group operations?

Side note: I had a little post-colloquium discussion with our fellow grad student Sam Lewallen, he asked:

Question: Is it possible for the same topological manifold to have two different Lie group structures where the induced smooth structures are different?

Note that neither the above nor Hilbert’s fifth problem shows such thing is impossible, since they both start with the phase ‘given a topological group’. My *guess* is this should be possible (so please let me know if you know the answer!) The first attempt might be trying to generate an exotic \mathbb{R}^4 from Lie group. Since the 3-dimensional Heisenberg group induces the standard (and unique) smooth structure on \mathbb{R}^3, I guess the 4-dimensional Heisenberg group won’t be exotic.

Anyways, so the Hilbert 5th problem was famously solved in the 50s by Montgomery-Zippin and Gleason, using set-theoretical methods (i.e. ultrafilters).

Gromov comes in later on and made the brilliant connection between (infinite) discrete groups and Lie groups. i.e. one see a discrete group as a metric space with word metric, ‘zoom out’ the space and produce a sequence of metric spaces, take the limit (Gromov-Hausdorff limit) and obtain a ‘continuous’ space. (which is ‘almost’ a Lie group in the sense that it’s an inverse limit of Lie groups.)

Hence he was able to adapt the machinery of Montgomery-Zippin to prove things about discrete groups:

Theorem: (Gromov) Any group with polynomial growth is virtually nilpotent.

Side note: I learned about this through the very detailed and well-presented course by Dave Gabai. (I thought I must have blogged about this, turns out I haven’t…)

The beauty of the theorem is (in my opinion) that we are given any discrete group, and all that’s known is how large the balls are (in fact, not even that, we know how large the large balls grow), yet the conclusion is all about the algebraic structure of the group. To learn more about Gromov’s work, see his paper. Although unrelated to the rest of this post, I shall also mention Bruce Kleiner’s paper where he proved Gromov’s theorem without using Hilbert’s 5th problem, instead he used space of harmonic maps on graphs.

Now we finally comes to a point of briefly mentioning the work of Tao et.al.! So they adopted Gromov’s methods of limiting and ‘ultra-filtering’ to apply to stuff that’s not even a whole discrete group: Since Gromov’s technique was to take the limit of a sequence of metric spaces which are zoomed out versions of balls in a group, but the Gromov-Hausdorff limit actually doesn’t care about the fact that those spaces are zoomed out from the same group, they may as well be just a family of subsets of groups with ‘bounded geometry’ of a certain kind.

Definition: An K-approximate group S is a (finite) subset of a group G where S\cdot S = \{ s_1 s_2 \ | \ s_1, s_2 \in S \} can be covered by K translates of S. i.e. there exists p_1, \cdots, p_K \in G where S \cdot S \subseteq \cup_{i=1}^k p_i \cdot S.

We shall be particularly interested in sequence of larger and larger sets (in cardinality) that are K-approximate groups with fixed K.

Examples:
Intervals [-N, N] \subseteq \mathbb{Z} are 2-approximate groups.

Balls of arbitrarily large radius in \mathbb{Z}^n are C \times 2^n approximate groups.

Balls of arbitrarily large radius in the 3-dimensional Heisenberg group are C \times 2^4 approximate groups. (For more about metric space properties of the Heisenberg group, see this post)

Just as in Gromov’s theorem, they started with any approximate group (a special case being sequence of balls in a group of polynomial growth), and concluded that they are in fact always essentially balls in Nilpotent groups. More precisely:

Theorem: (Breuillard-Green-Tao) Any K-approximate group S in G is covered by C(K) many translates of subgroup G_0 < G where G_0 has a finite (depending only on K) index nilpotent normal subgroup N.

With this theorem they were able to re-prove (see p71 of their paper) Cheeger-Colding’s result that

Theorem: Any closed n dimensional manifold with diameter 1 and Ricci curvature bounded below by a small negative number depending on n must have virtually nilpotent fundamental group.

Where Gromov’s theorem yields the same conclusion only for non-negative Ricci curvature.

Random thoughts:

1. Can Kleiner’s property T and harmonic maps machinery also be used to prove things about approximate groups?

2. The covering definition as we gave above in fact does not require approximate group S to be finite. Is there a Lie group version of the approximate groups? (i.e. we may take compact subsets of a Lie group where the self-product can be covered by K many translates of the set.) I wonder what conclusions can we expect for a family of non-discrete approximate groups.

As promised, I shall say a few words about the Hilbert-Smith conjecture and drop a note on the recent proof of it’s 3-dimensional case by Pardon.

From the solution of Hilbert’s fifth problem we know that any topological group that is a n-manifold is automatically equipped with a smooth structure compatible with group operations. What if we don’t know it’s a manifold? Well, of course then they don’t have to be a Lie group, for example the p-adic integer group \mathbb{Z}_p is homeomorphic to a Cantor set hence is not a Lie group. Hence it makes more sense to ask:

Hilbert-Smith conjecture: Any topological group acting faithfully on a connected n-manifold is a Lie group.

Recall an action is faithful if the homomorphism \varphi: G \rightarrow homeo(M) is injective.

As mentioned in Tao’s post, in fact \mathbb{Z}_p is the only possible bad case! i.e. it is sufficient to prove

Conjecture: \mathbb{Z}_p cannot act faithfully on a finite dimensional connected manifold.

The exciting new result of Pardon is that by adapting 3-manifold techniques (finding incompressible surfaces and induce homomorphism to mapping class groups) he was able to show:

Theorem: (Pardon ’12) There is no faithful action of \mathbb{Z}_p on any connected 3-manifolds.

And hence induce the Hilbert-Smith conjecture for dimension 3.

Discovering this result a few days ago has been quite exciting, I would hope to find time reading and blogging about that in more detail soon.

Cutting the Knot

Conan10 Comments

Recently I came across a paper by John Pardon – a senior undergrad here at Princeton; in which he answered a question by Gromov regarding “knot distortion”. I found the paper being pretty cool, hence I wish to highlight the ideas here and perhaps give a more pictorial exposition.

This version is a bit different from one in the paper and is the improved version he had after taking some suggestions from professor Gabai. (and the bound was improved to a linear one)

Definition: Given a rectifiable Jordan curve \gamma: S^1 \rightarrow \mathbb{R}^3, the distortion of \gamma is defined as

\displaystyle \mbox{dist}(\gamma) = \sup_{t,s \in S^1} \frac{d_{S^1}(s,t)}{d_{\mathbb{R}^3}(\gamma(s), \gamma(t))}.

i.e. the maximum ratio between distance on the curve and the distance after embedding. Indeed one should think of this as measuring how much the embedding ‘distort’ the metric.

Given knot \kappa, define the distortion of \kappa to be the infimum of distortion over all possible embedding of \gamma:

\mbox{dist}(\kappa) = \inf\{ \mbox{dist}(\gamma) \ | \ \gamma \ \mbox{is an embedding of} \ \kappa \ \mbox{in} \ \mathbb{R}^3 \}

It was (somewhat surprisingly) an open problem whether there exists knots with arbitrarily large distortion.

Question: (Gromov ’83) Does there exist a sequence of knots (\kappa_n) where \lim_{n \rightarrow \infty} \mbox{dist}(\kappa_n) = \infty?

Now comes the main result in the paper: (In fact he proved a more general version with knots on genus g surfaces, for simplicity of notation I would focus only on torus knots)

Theorem: (Pardon) For the torus knot T_{p,q}, we have

\mbox{dist}(T_{p,q}) \geq \frac{1}{100} \min \{p,q \}

.

To prove this, let’s make a few observations first:

First, fix a standard embedding of \mathbb{T}^2 in \mathbb{R}^3 (say the surface obtained by rotating the unit circle centered at (2, 0, 0) around the z-axis:

and we shall consider the knot that evenly warps around the standard torus the ‘standard T_{p,q} knot’ (here’s what the ‘standard T_{5,3} knot looks like:

By definition, an ’embedding of the knot’, is a homeomorphism \varphi:\mathbb{R}^3 \rightarrow \mathbb{R}^3 that carries the standard T_{p,q} to the ‘distorted knot’. Hence the knot will lie on the image of the torus (perhaps badly distorted):

For the rest of the post, we denote \varphi(T_{p,q}) by \kappa and \varphi(\mathbb{T}^2) by T^2, w.l.o.g. we also suppose p<q.

Definition: A set S \in T^2 is inessential if it contains no homotopically non-trivial loop on T^2.

Some important facts:

Fact 1: Any homotopically non-trivial loop on \mathbb{T}^2 that bounds a disc disjoint from T^2 intersects T_{p,q} at least p times. (hence the same holds for the embedded copy (T^2, \kappa)).

As an example, here’s what happens to the two generators of \pi_1(\mathbb{T}^2) (they have at least p and q intersections with T_{p,q} respectively:

From there we should expect all loops to have at least that many intersections.

Fact 2: For any curve \gamma and any cylinder set C = U \times [z_1, z_2] where U is in the (x,y)-plane, let U_z = U \times \{z\} we have:

\ell(\gamma \cap C) \geq \int_{z_1}^{z_2} | \gamma \cap U_z | dz

i.e. The length of a curve in the cylinder set is at least the integral over z-axis of the intersection number with the level-discs.

This is merely saying the curve is longer than its ‘vertical variation’:

Similarly, by considering variation in the radial direction, we also have

\ell(\gamma \cap B(\bar{0}, R) \geq \int_0^{R} | \gamma \cap \partial B(\bar{0}, r) | dr

Proof of the theorem

Now suppose \mbox{dist}(T_{p,q})<\frac{1}{100}p, we find an embedding (T^2, \kappa) with \mbox{dist}(\kappa)<\frac{1}{100}p.

For any point x \in \mathbb{R}^3, let

\rho(x) = \inf \{ r \ | \ T^2 \cap (B(x, r))^c is inessential \}

i.e. one should consider \rho(x) as the smallest radius around x so that the whole ‘genus’ of T^2 lies in B(x,\rho(x)).

It’s easy to see that \rho is a positive Lipschitz function on \mathbb{R}^3 that blows up at infinity. Hence the minimum value is achieved. Pick x_0 \in \mathbb{R}^3 where \rho is minimized.

Rescale the whole (T^2, \kappa) so that x_0 is at the origin and \rho(x_0) = 1.

Since \mbox{dist}(\kappa) < \frac{1}{100}p (and note distortion is invariant under scaling), we have

\ell(\kappa \cap B(\bar{0}, 1) < \frac{1}{100}p \times 2 = \frac{1}{50}p

Hence by fact 2, \int_1^{\frac{11}{10}} | \kappa \cap \partial B( \bar{0}, r)| dr \leq \ell(\kappa \cap B(\bar{0}, 1)) < \frac{1}{40}p

i.e. There exists R \in [1, \frac{11}{10}] where the intersection number is less or equal to the average. i.e. | \kappa \cap \partial B(\bar{0}, R) | \leq \frac{1}{4}p

We will drive a contradiction by showing there exists x with \rho(x) < 1.

Let C_z = B(\bar{0},R) \cap \{z \in [-\frac{1}{10}, \frac{1}{10}] \}, since

\int_{-\frac{1}{10}}^{\frac{1}{10}} | U_t \cap \kappa | dt \leq \ell(\kappa \cap B(\bar{0},1) ) < \frac{1}{50}p

By fact 2, there exists z_0 \in [-\frac{1}{10}, \frac{1}{10}] s.t. | \kappa \cap B(\bar{0},1) \times \{z_0\} | < \frac{1}{10}p. As in the pervious post, we call B(\bar{0},1) \times \{z_0\} a ‘neck’ and the solid upper and lower ‘hemispheres’ separated by the neck are U_N, U_S.

Claim: One of U_N^c \cap T^2, \ U_S^c \cap T^2 is inessential.

Proof: We now construct a ‘cutting homotopy’ h_t of the sphere S^2 = \partial B(\bar{0}, R):

i.e. for each t \in [0,1), \ h_t(S^2) is a sphere; at t=1 it splits to two spheres. (the space between the upper and lower halves is only there for easier visualization)

Note that during the whole process the intersection number h_t(S^2) \cap \kappa is monotonically increasing. Since | \kappa \cap B(\bar{0},R) \times \{z_0\} | < \frac{1}{10}p, it increases no more than \frac{1}{5}p.

Observe that under such ‘cutting homotopy’, \mbox{ext}(S^2) \cap T^2 is inessential then \mbox{ext}(h_1(S^2)) \cap T^2 is also inessential. (to ‘cut through the genus’ requires at least p many intersections at some stage of the cutting process, but we have less than \frac{p}{4}+\frac{p}{5} < \frac{p}{2} many interesections)

Since h_1(S^2) is disconnected, the ‘genus’ can only lie in one of the spheres, we have one of U_N^c \cap T^2, \ U_S^c \cap T^2 is inessential. Establishes the claim.

We now apply the process again to the ‘essential’ hemisphere to find a neck in the ydirection, i.e.cutting the hemisphere in half in (x,z) direction, then the (y,z)-direction:

The last cutting homotopy has at most \frac{p}{5} + 3 \times \frac{p}{4} < p many intersections, hence has inessential complement.

Hence at the end we have an approximate \frac{1}{8} ball with each side having length at most \frac{6}{5}, this shape certainly lies inside some ball of radius \frac{9}{10}.

Let the center of the \frac{9}{10}-ball be x. Since the complement of the \frac{1}{8} ball intersects T^2 in an inessential set, we have B(x, \frac{9}{10})^c \cap T^2 is inessential. i.e.

\rho(x) \leq \frac{9}{10} <1

Contradiction.