Well, part of the conclusion of Theorem 1 is that smooth structure placed on G is not just any old smooth structure – it is the one inherited from the linear group by locally pulling back under the faithful representing map (because is locally , and is locally smooth and non-degenerate). One corollary of this is that if H is any closed subgroup of G (and hence also locally compact, with being a sub-Lie algebra of ), then H will indeed be a smooth manifold of G, with the smooth structure of H being the restriction of the smooth structure of G.

Applying this to the connected group and the subgroup that is the graph of the outer automorphism (both of which are faithfully represented by the same representation ), we see that this graph is indeed a smooth subgroup.

In any event, the argument that gives that any outer automorphism is smooth is also the argument used to give Remark 3, which implies in particular that Lie groups have a unique smooth structure.

]]>in order to show the analyticity of the outer automorphism on you use the connected case of Theorem 1 for its graph, a closed connected subgroup of .

Does this step require uniqueness of the smooth structure provided by Theorem 1? The problem is that one needs the graph to be an analytic *sub*manifold of , as opposed to admitting an arbitrary analytic manifold structure.

regards, pavel

]]>5.4. Proposition. For a locally compact group G the following assertions are

equivalent:

(i) G admits a contractive automorphism group;

(ii) G is a simply connected Lie group whose Lie algebra g admits a positive

graduation.

The corresponding result for local groups is proved in L. van den Dries, I. Goldbring, Locally Compact Contractive Local Groups, arXiv:0909.4565v2.

I used Siebert result for proving the Lie algebraic structure of the metric tangent space to a sub-riemannian manifold in M. Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136. arXiv:0804.0135v2

When saying that contractive automorphisms, or approximately contractive automorphisms, may be more relevant than one-parameter subgroups, I am thinking about sub-riemannian geometry again, where a one-parameter subgroup of a group, endowed with a left-invariant distribution and a corresponding Carnot-Caratheodory distance, is “smooth” (with respect to

Pansu-type derivative) if and only if the generator is in the distribution. Metrically speaking, if the generator is not in the distribution then any trajectory of the one-parameter group has Hausdorff dimension greater than one. That means lots of problems with the definition of the exponential and any reasoning based on differentiating flows.