By Valentino Magnani

We receive an intrinsic Blow-up Theorem for normal hypersurfaces on graded nilpotent teams. This process permits us to symbolize explicitly the Riemannian floor degree when it comes to the round Hausdorff degree with recognize to an intrinsic distance of the crowd, particularly homogeneous distance. We observe this consequence to get a model of the Riemannian coarea forumula on sub-Riemannian teams, that may be expressed by way of arbitrary homogeneous distances.We introduce the average type 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. through a similar 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 end up that it has finite Q − 2 Hausdorff degree, the place Q is the homogeneous size of the gang.

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This rules out the case e = 2. HwrG IG. G S; S3wr Z5 a~ld conclusion (iii) holds. We can thus assume that V is For now) assume that, G and By our hypotheses, G has an ~be1iall Sylow 2-subgroup. F however has 2 p = 2 = m. Then 3 (cf. 13), G S; f(3 )wrZ2 and 'conclusion (v) holds. The remailiing case iS,when m I lUI, FIT of order 22. Consequently, A/F ~ S3 orZ3, and it follows that p = 2. {p, q} = {2, 3}. In' all such cases, GIG is a {2,3}-group. Thus IGI must be divisible by T (1' ~ 5), and so there exists a This only occurs in the /= p-group, we must have that G IA is a 2-group.

Dl(G) ~ ~ log3(n). (b) Let V =I=- 0 be a faithful and completely reducible F[G]-modulc over an arbitrary field F. r(V). Then dl( G) ~,8 -I- ~ log3 (n/8). Proof. e. mong all counterexamples to the theorem choose one with IGln minimal. For x > 0, we let a(x)'= ~ log3(x) and f3(x) = 8 + ~ log3(x/8) = a(x) -1-:3 - ~ 10g3(8). Proof. r(W*) = 4 . r(W) and IH* I = 241HI4 ~ Observe that if IHI = 11Vla/ A, then IH*I = 24· (11Vl aI A)4 = (24/A 3;)IWI 4a l).. , Of course, H* is solvable if alld only if His.

Since O'(x) is increasing, we have that dI(G) ~ O'(pt) = a(n) unless 3 18 / 5 [/8 loss of generality to assume that :F is algebraically closed. 15 or dl(GL(l,p» pi rnax dl( G / M). max dl(G) [a(pl)] 2 0 1 1 3 or 5 22 1 2 2 or 3. 2' 3 3 32 4 5 5 23 24 2 (Cor. 2: 13) 3 4 4 (Cor. 15) 5 6 Iil all cases, dl(G)::; a(pl) = 3 nj > 0, then each Vi is By induction, we conclude ~ max{(3(nj)li = 1,2} ~ = 1,2} (3(n). ~ G maximal such that Ve is not homogeneous and write Ve = VI ED ... EB Vt for homogeneous components Vi of Ve, t > 1.

### A Blow-up Theorem for regular hypersurfaces on nilpotent groups by Valentino Magnani

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