# Escape property of the Gleason metric and sub-riemannian distances again

The last post of Tao from his series of posts on the Hilbert’s fifth problem contains interesting results which can be used for understanding the differences between Gleason distances and sub-riemannian distances or, more general, norms on groups with dilations.

For normed groups with dilations see my previous post (where links to articles are also provided). Check my homepage for more details (finally I am online again).

There is also another post of mine on the Gleason metric (distance) and the CC (or sub-riemannian) distance, where I explain why the commutator estimate (definition 3, relation (2) from the last post of Tao) forces “commutativity”, in the sense that a sub-riemannian left invariant distance on a Lie group which has the commutator estimate must be a riemannian distance.

What about the escape property (Definition 3, relation (1) from the post of Tao)?

From his Proposition 10 we see that the escape property implies the commutator estimate, therefore a sub-riemannian left invariant distance with the escape property must be riemannian.

An explanation of this phenomenon can be deduced by using the notion of “coherent projection”, section 9 of the paper

A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111

in the very particular case of sub-riemannian Lie groups (or for that matter normed groups with dilations).

Suppose we have a normed group with dilations $(G, \delta)$ which has another left invariant dilation structure on it (in the paper this is denoted by a “$\delta$ bar”, here I shall use the notation $\alpha$ for this supplementary dilation structure).

There is one such a dilation structure available for any Lie group (notice that I am not trying to give a proof of the H5 problem), namely for any $\varepsilon > 0$ (but not too big)

$\alpha_{\varepsilon} g = \exp ( \varepsilon \log (g))$

(maybe interesting: which famous lemma is equivalent with the fact that $(G,\alpha)$ is a group with dilations?)
Take $\delta$ to be a dilation structure coming from a left-invariant distribution on the group . Then $\delta$ commutes with $\alpha$ and moreover

(*) $\lim_{\varepsilon \rightarrow 0} \alpha_{\varepsilon}^{-1} \delta_{\varepsilon} x = Q(x)$

where $Q$ is a projection: $Q(Q(x)) = x$ for any $x \in G$.

It is straightforward to check that (the left-translation of) $Q$ (over the whole group) is a coherent projection, more precisely it is the projection on the distribution!

Exercise: denote by $\varepsilon = 1/n$ and use (*) to prove that the escape property of Tao implies that $Q$ is (locally) injective. This implies in turn that $Q = id$, therefore the distribution is the tangent bundle, therefore the distance is riemannian!

UPDATE:    See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for understanding in detail the role of the escape property in the proof of the Hilbert 5th problem.

# Gleason metric and CC distance

In the series of posts on Hilbert’s fifth problem, Terence Tao defines a Gleason metric, definition 4 here, which is a very important ingredient of the proof of the solution to H5 problem.

Here is Remark 1. from the post:

The escape and commutator properties are meant to capture “Euclidean-like” structure of the group. Other metrics, such as Carnot-Carathéodory metrics on Carnot Lie groups such as the Heisenberg group, usually fail one or both of these properties.

I want to explain why this is true. Look at the proof of theorem 7. The problem comes from the commutator estimate (1). I shall reproduce the relevant part of the proof because I don’t yet know how to write good-looking latex posts:

From the commutator estimate (1) and the triangle inequality we also obtain a conjugation estimate

$\displaystyle \| ghg^{-1} \| \sim \|h\|$

whenever ${\|g\|, \|h\| \leq \epsilon}$. Since left-invariance gives

$\displaystyle d(g,h) = \| g^{-1} h \|$

we then conclude an approximate right invariance

$\displaystyle d(gk,hk) \sim d(g,h)$

whenever ${\|g\|, \|h\|, \|k\| \leq \epsilon}$.

The conclusion is that the right translations in the group are Lipschitz (with respect to the Gleason metric). Because this distance (I use “distance” instead of “metric”) is also left invariant, it follows that left and right translations are Lipschitz.

Let now G be a connected Lie group with a left-invariant distribution, obtained by left translates of a vector space D included in the Lie algebra of G. The distribution is completely non-integrable if D generates the Lie algebra by using the + and Lie bracket operations. We put an euclidean norm on D and we get a CC distance on the group defined by: the CC distance between two elements of the group equals the infimum of lengths of horizontal (a.e. derivable, with the tangent in the distribution) curves joining the said points.

The remark 1 of Tao is a consequence of the following fact: if the CC distance is right invariant then D equals the Lie algebra of the group, therefore the distance is riemannian.

Here is why: in a sub-riemannian group (that is a group with a distribution and CC distance as explained previously) the left translations are Lipschitz (they are isometries) but not all right translations are Lipschitz, unless D equals the Lie algebra of G. Indeed, let us suppose that all right translations are Lipschitz. Then, by Margulis-Mostow version (see also this) of the Rademacher theorem , the right translation by an element “a” is Pansu derivable almost everywhere. It follows that the Pansu derivative of the right translation by “a” (in almost every point) preserves the distribution. A simple calculus based on invariance (truly, some explanations are needed here) shows that by consequence the adjoint action of “a” preserves D. Because “a” is arbitrary, this implies that D is an ideal of the Lie algebra. But D generates the Lie algebra, therefore D equals the Lie algebra of G.

If you know a shorter proof please let me know.

UPDATE: See the recent post 254A, Notes 4: Bulding metrics on groups, and the Gleason-Yamabe theorem by Terence Tao, for details of the role of the Gleason metric  in the proof of the Hilbert 5th problem.