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.

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