By Valentino Magnani

We receive an intrinsic Blow-up Theorem for normal hypersurfaces on graded nilpotent teams. This strategy permits us to symbolize explicitly the Riemannian floor degree by way of the round Hausdorff degree with appreciate to an intrinsic distance of the gang, particularly homogeneous distance. We observe this end result to get a model of the Riemannian coarea forumula on sub-Riemannian teams, that may be expressed when it comes to arbitrary homogeneous distances.We introduce the typical category of horizontal isometries in sub-Riemannian teams, giving examples of rotational invariant homogeneous distances and rotational teams, the place the coarea formulation takes a less complicated shape. by way of an identical Blow-up Theorem we receive an optimum estimate for the Hausdorff measurement of the attribute set relative to C1,1 hypersurfaces in 2-step teams and we turn out that it has finite Q − 2 Hausdorff degree, the place Q is the homogeneous size of the crowd.

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32]). i E, and H of a free ~roup (G:H) rank G then - 1) + 1. and let Then the maximal are finite G Y be the bouquet subtree T of Y is 37 THE TRIVIAL just the u n i q u e v e r t e x of CASE Y, and G = n(Y,T) so acts f r e e l y on the tree X = F(Y,T). 1. 8 is (G:H) trivial of [E(H\X) I = (G:H)IE(G\X) I : (G:H)(rank (Bass-Serre, srouP, and H G\X, - G - + acts is finite so if rank freely then G is also 1 (G:H) I V ( G \ X ) I - 1) + 1. + 1 D of C h a p t e r and in fact Serre [77]). i we get a very Let any subgroup of with each c o n j q s a t e p~(v)p (p e z(~,T), ~:Y + z(~,T) be which has of the image of each v E V(Y)).

7 index can graph whose order. generalization (Serre If [77]). (Gi'F) .. 2. THEOREM Proof. is that, group of a c o n n e c t e d are finite of bounded Let us note the f o l l o w i n g formula, and c l e a r l y Since F is i s o m o r p h i c the n u m b e r : subgroup l E(Y) of F\X IF\G/Gel _ subgroup H G. 1. G:r) [ IGel ' E(Y) seen s t a t e m e n t s of of finite V(Y) = for vertices. subgroups and the vertex groups is Now gives the desired result. We have F 1 [Gel to the f u n d a m e n t a l E(Y) and s i m i l a r l y is finite does not meet any conjugate of edges [ Y More G.