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

By Valentino Magnani

We receive an intrinsic Blow-up Theorem for normal hypersurfaces on graded nilpotent teams. This technique permits us to symbolize explicitly the Riemannian floor degree by way of the round Hausdorff degree with recognize to an intrinsic distance of the crowd, specifically homogeneous distance. We follow this outcome 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 ordinary 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. via 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 measurement of the crowd.

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Critical densities for various properties A bunch of properties are now known to hold for random groups. This ranges from group combinatorics (small cancellation properties) to algebra (freeness of subgroups) to geometry (boundary at infinity, growth exponent, CAT(0)-ness) to probability (random walk in the group) to representation theory on the Hilbert space (property (T ), Haagerup property). Some of the properties studied here are intrinsic to the group, others depend on a marked set of generators or on the standard presentation through which the random group was obtained.

But they probably extend to small positive values of d, the determination of which is an interesting problem. The first such theorem [AO96] is the following: Theorem 17 – With overwhelming probability in the few-relator model of random groups with m generators and n relators, any subgroup generated by m − 1 elements is free. ). When the m − 1 generators of the subgroup are chosen among the m standard generators of the group, this is a particular case of Theorem 6. The group itself is not free and more precisely [Ar97]: Theorem 18 – In the few-relator model of random groups with n relators, no finite-index subgroup of the group is free.

Let 0 d 1 and let W be the set of all (2m) words of length in ’s. Let R be the random set obtained by picking (2m)d times at the a±1 i random a word in W . Typical elements in a group 51 Let G = G0 / R be the random quotient so obtained. Then with overwhelming probability: • If d < dcrit , then G is (non-elementary) hyperbolic. • If d > dcrit , then G = {e}. Once again the spirit of the density model is to kill a number of words equal to some power d of the total number of words (2m) . Note that dcrit < 1/2, even when G0 is the free group (the difference with Theorem 11 being that we use plain words instead of reduced ones).

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