ON TRANSVERSAL SUBMANIFOLDS AND THEIR MEASURE

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1 ON TRANSVERSAL SUBMANIFOLDS AND THEIR MEASURE VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE Abstract: We study the class of transversal submanifolds in Carnot groups. We characterize their blow-ups at transversal points and prove a negligibility theorem for their generalized characteristic set, with respect to the Carnot-Carathéodory Hausdorff measure. This set is made by all points of non-maximal degree. Observing that C 1 submanifolds in Carnot groups are generically transversal, the previous results prove that the intrinsic measure of C 1 submanifolds is generically equivalent to their Carnot-Carathéodory Hausdorff measure. As a result, the restriction of this Hausdorff measure to the submanifold can be replaced by a more manageable integral formula, that should be seen as a sub-riemannian mass. Another consequence of these results is an explicit formula, only depending on the embedding of the submanifold, that computes the Carnot-Carathéodory Hausdorff dimension of C 1 transversal submanifolds. Contents 1. Introduction 2 2. Notation and preliminary results 5 3. Blow-up at transversal points Negligibility of lower degree points in transversal submanifolds Size of the characteristic set for C 1,λ submanifolds 26 References 28 Date: July 10, Mathematics Subject Classification. Primary 53C17; Secondary 22E25, 28A78. Key words and phrases. Stratified groups, submanifolds, Hausdorff measure. The first author acknowledges the support of the European Project ERC AdG *GeMeThNES*. The second author acknowledges the support of the US National Science Foundation Grants DMS and DMS The third author acknowledges the support of MIUR, GNAMPA of INDAM (Italy), University of Padova, Fondazione CaRiPaRo Project Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems and University of Padova research project Some analytic and differential geometric aspects in Nonlinear Control Theory, with applications to Mechanics. 1

2 2 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE 1. Introduction A stratified group G is a connected, simply connected nilpotent Lie group, whose Lie algebra G has a special grading that allows for the existence of natural dilations along with a homogeneous distance that respect both dilations and group operation. A stratified group equipped with a sub-riemannian distance is also known as Carnot group. The first developments of Geometric Analysis in Carnot groups were mainly focused on geometric properties of domains in relation with Sobolev embeddings (see, for instance, [6], [12], [16]), problems from the calculus of variations (e.g., [5], [10]), differential geometric calculus on hypersurfaces ([9]), and the structure of finite perimeter sets (a very incomplete list of references includes [1], [7], [8], [13], [14], and [20]). The preceding lists of references are far from exhaustive, representing only a small sample of the rapidly expanding literature in the field of sub-riemannian geometric analysis. The study of finite perimeter sets and domains naturally connects with the study of hypersurfaces and their Hausdorff measure. The fact that this measure is constructed by a fixed homogeneous distance of the group is understood. An important object in this context is the so-called G-perimeter measure. It can be defined using a volume measure and a smooth left invariant metric on the horizontal subbundle of the group. This measure is equivalent, in the sense of (1.2) below, to the (Q 1)-dimensional Hausdorff measure either of the reduced boundaries in step two Carnot groups, [14], or of the topological boundaries of C 1 smooth domains in arbitrary stratified groups, [20], where Q is the Hausdorff dimension of G. The G-perimeter measure for regular sets has a precise integral formula that replaces the Hausdorff measure and that does not contain the homogeneous distance. In fact, it is more manageable for minimization problems. In the development of Geometric Measure Theory on stratified groups, a natural question arises: what is the right measure replacing the G-perimeter measure for higher codimensional sets? In [23], a general integral formula for the intrinsic measure of C 1 submanifolds has been found: let Σ be a C 1 smooth submanifold of G and define ˆ (1.1) µ Σ (U) = ( t1 Φ tp Φ) D,Φ(t) dt Φ 1 (U) where Φ : A U Σ is a local parametrization of Σ, A is an open set of R p and D is the degree of Σ, see Subsection 2.4 for more details. This measure yields the perimeter measure in codimension one and in several cases it is equivalent to H D Σ up to geometric constants, namely, (1.2) C 1 H D Σ µ Σ C H D Σ. We stress the fact that the degree D of Σ is also equal to the Hausdorff dimension of Σ. This equivalence already appears in [23] for C 1,1 smooth submanifolds in stratified groups, under the key assumption that points of degree less than D are H D -negligible. Under this assumption, the equivalence (1.2) is a consequence of a blow-up theorem performed at each point of degree D, see [23, Theorem 1.1]. For more details on the notion of degree, see Subsection 2.4.

3 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 3 The previously mentioned H D -negligibility condition holds in many cases: for C 1,1 smooth submanifolds in two step stratified groups [22], and in the Engel group [19], for C 1 smooth non-horizontal submanifolds in all stratified groups [20, 21], and for C 1 smooth curves in all groups [18]. In all these cases, the equivalence (1.2) holds. In fact, when Σ is C 1,1 this is a consequence of the blow-up theorem of [23], while for the case of C 1 smoothness the blow-up at points of degree D is established in [21] for non-horizontal submanifolds and in [18] for all curves. Surprisingly, for C 1 smooth submanifolds in stratified groups the equivalence (1.2) is an intriguing open question. One of the reasons behind this new difficulty is that, in higher codimension, submanifolds may belong to different classes, namely, they may have different Hausdorff dimensions, while keeping the same topological dimension. Simple examples of this phenomenon are given by the one dimensional homogeneous subgroups, that have different Hausdorff dimensions according to their degree. Clearly, analogous examples can be easily found for higher dimensional homogeneous subgroups. It is instructive to compare these cases with that of codimension one submanifolds, whose Hausdorff dimension must equal Q 1. Can we detect the right class of submanifolds that has the good behaviour of hypersurfaces, and replaces them in higher codimension? When the codimension is low, precisely less than the dimension of the horizontal fibers, this class is formed by non-horizontal submanifolds, for which (1.2) holds, [21]. In higher codimension, this class is formed by transversal submanifolds. A transversal submanifold is easily defined as a top-dimensional submanifold among all submanifolds having the same topological dimension p. We have a precise formula for this maximal Hausdorff dimension D(p), see Section 2 for precise definitions. In this paper, we prove that transversal submanifolds in arbitrary codimension have properties similar to those of hypersurfaces. In fact, our main result is that (1.2) holds for all C 1 smooth transversal submanifolds in arbitrary stratified groups. This follows by combining two key results: a blow-up theorem and a negligibility result, that are stated below. The estimates (1.2) show in particular that the Hausdorff dimension of C 1 smooth transversal submanifolds is equal to D(p). This fact should be compared with [17, 0.6.B], where M. Gromov provides a formula for the Hausdorff dimension of generic submanifolds. Gromov also introduces the number D H (Σ) associated with a submanifold Σ; this number coincides with the degree d(σ) introduced in Subsection 2.4, see [22, Remark 2] for a proof of this fact. Another motivation for our study of transversal submanifolds is that C 1 smooth submanifolds are generically transversal, namely, most C 1 submanifolds are transversal. This suggests that these submanifolds are important in the subsequent study of higher codimensional submanifolds in Carnot groups. The fact that transversality is a generic property can be seen for instance as a simple consequence of our Lemma 2.11 and then arguing as in [21, Section 4]. The main results of our work are a blow-up theorem and an H D(p) -negligibility theorem for all C 1 smooth transversal submanifolds. These theorems extend the blowup theorem of [21] and the negligibility theorem of [20].

4 4 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE Theorem 1.1 (Blow-up theorem). If Σ is a C 1 smooth submanifold of topological dimension p and x Σ is transversal, then, for every compact neighbourhood F of 0, we have (1.3) F δ 1/r (x 1 Σ) F Π Σ (x) as r 0 + where the convergence is in the sense of the Hausdorff distance between compact sets and Π Σ (x) is a p-dimensional normal homogeneous subgroup of G, having Hausdorff dimension equal to D(p). Moreover, the following limit holds ( ) µ Σ B(x, r) (1.4) lim r 0 + r D(p) g(( = θd τσ (x) ) ) D(p),x ( τ Σ, g (x) ). D(p),x The proof of this theorem is given in Section 3. This section, along with Section 2, also contains the definitions of the relevant notions. It is worth to mention that in the case of C 1,1 regularity the blow-up at transversal points is already contained in [23]. In our case, where Σ is only C 1, the approach of [23] does not apply, so we follow the method used in [18] for curves. The point here is to provide a special weighted reparametrization of Σ around the blow-up point, see (3.8). Our next main result is the following generalized negligibility theorem. Theorem 1.2 (Negligibility theorem). Let Σ G be a p-dimensional C 1 submanifold and let Σ c Σ denote the subset of points with degree less than D(p). We have (1.5) H D(p) (Σ c ) = 0. We refer to Section 4 for the definition of the generalized characteristic set Σ c. The proof of Theorem 1.2 relies on covering arguments and a number of technical lemmata, that aim to estimate the behaviour of the number of small balls covering the generalized characteristic set. The difficulty here is to properly translate the information on the lower degree of the points into concrete estimates on the best local coverings around these points, see Lemma 2.3, Lemma 4.1 and Lemma 4.2. Theorem 1.2 extends to Euclidean Lipschitz submanifolds using standard arguments (see Theorem 4.5); more precisely, for any p-dimensional Lipschitz regular submanifold Σ G there holds H D(p) (Σ Σ c ) = 0, where Σ is the set of points where the (Euclidean) tangent cone to Σ is not a p- dimensional plane, see Subsection 2.4. Arguing the same way, one also realizes that estimates (1.2) extend to all Euclidean Lipschitz transversal submanifolds. The method to prove Theorem 1.2 can also be used to establish new estimates on the Carnot-Carathéodory Hausdorff dimension of Σ c for C 1,λ submanifolds in Carnot groups, where 0 < λ 1. These estimates show that the Carnot-Carathéodory Hausdorff dimension of Σ c can be estimated from above by a bound smaller than D(p). More precisely, we have

5 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 5 Theorem 1.3. Let Σ be a p-dimensional submanifold of G of class C 1,λ, λ (0, 1]. Then dim H Σ c D(p) λ if l = l(p) = 1 (1.6) dim H Σ c D(p) 1 if l 2 and λ 1 l 1. dim H Σ c D(p) lλ if l 2 and λ 1 1+λ l 1 Both Theorems 1.2 and 1.3 generalize some results proved, in the Heisenberg group framework, in the fundamental paper [2], compare Remark 5.3. We do not know whether the estimates of Theorem 1.3 are sharp. Even in Heisenberg groups, this sharpness seems to be an interesting open question. We refer to [3] for results and open problems akin to that of estimating the size of Σ c. The validity of (1.2) for a large class of submanifolds makes the intrinsic measure (1.1) a reasonable notion of sub-riemannian mass. This should be seen for instance in the perspective of studying special classes of isoperimetric inequalities, when either the filling current or the filling submanifold must be necessarily transversal, as it occurs for higher dimensional fillings in Heisenberg groups. 2. Notation and preliminary results 2.1. Carnot groups and exponential coordinates. Let us start with a brief introduction to stratified groups; we refer to [15] for more details on the subject. Let G be a connected and simply connected Lie group with stratified Lie algebra G = V 1 V ι of step ι, satisfying the conditions V i+1 = [V 1, V i ] for every i 1 and V ι+1 = {0}. We set (2.1) n j := dim V j and m j := n n j, j = 1,..., ι ; we will also use m 0 := 0. The degree d j of j {1,..., n} is defined by the condition m dj j m dj. We denote by n the dimension of G, therefore n = m ι. We say that a basis (X 1,..., X n ) of G is adapted to the stratification, or in short adapted, if X mj 1 +1,..., X mj is a basis of V j for any j = 1,..., ι. In the sequel, we will fix a graded metric g on G, namely, a left invariant Riemannian metric on G such that the subspaces V k are orthogonal. Definition 2.1. An adapted basis (X 1,..., X n ) of G that is also orthonormal with respect to a left invariant Riemannian metric is a graded basis. Clearly, the Riemannian metric in the previous definition must be necessarily graded. When either an adapted or a graded basis is understood, we identify G with R n by the corresponding exponential coordinates of the first kind. We use two different ways of denoting points x of G with respect to fixed exponential coordinates of the first kind adapted to a graded basis of G. We use both the standard notation with lower indices x = (x 1,..., x n ) R n

6 6 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE and the one with upper indices x = (x 1, x 2,..., x ι ) R n 1 R n 2 R nι, where clearly x j = (x mj 1 +1,..., x mj ) R n j for all j = 1,..., ι. By the Baker- Campbell-Hausdorff formula, the group law reads in coordinates as (2.2) x y = x + y + Q(x, y), for a suitable polynomial function Q : R n R n R n. Precisely, Q = (Q 1,..., Q n ) can be written in the form (2.3) Q j (x, y) = R kh j (x, y)(x k y h x h y k ) j = 1,..., n k,h:d k <d j,d h <d j for suitable polynomial functions R kl j. It follows that for any bounded set K G there exists C = C(K) > 0 such that (2.4) Q(x, y) C x y for any x, y K Stratified groups as abstract vector spaces. To emphasize some intrinsic notions on stratified groups, while preserving the ease of using a linear structure, stratified groups can be also regarded as abstract vector spaces. In fact, connected and simply connected nilpotent Lie groups are diffeomorphic to their Lie algebra through the exponential mapping exp : G G, that is an analytic diffeomorphism. As a result, we equip the Lie algebra of G with a Lie group operation, given by the Baker-Campbell- Hausdorff series, that makes this Lie group isomorphic to the original G. This allows us to consider G as an abstract linear space, equipped with a polynomial operation and a grading G = H 1 H ι. Under this identification a graded basis (X 1,..., X n ) becomes an orthonormal basis of G as a vector space, where (X mj 1 +1,..., X mj ) is an orthonormal basis of H j for all j = 1,..., ι. Remark 2.2. When a stratified group G is seen as an abstract vector space, equipped with a graded basis X 1,..., X n, then the associated graded metric g makes this basis orthonormal. As a result, the metric g becomes the Euclidean metric with respect to the corresponding coordinates (x 1, x 2,..., x ι ) Dilations. For every r > 0, a natural group automorphism δ r : G G can be defined as the unique algebra homomorphism such that δ r (X) := rx for every X V 1. This one parameter group of linear isomorphisms constitutes the family of the so-called dilations of G. They canonically yield a one parameter group of dilations on G and can be denoted by the same symbol. With respect to our coordinates, we have δ r (x 1,..., x j,..., x n ) = (rx 1,..., r d j x j,..., r ι x n ) Left translations. For each element x G, the group operation of G automatically defines the corresponding left translation l x : G G, with l x (z) = xz for all z G. Right translations r x are defined in analogous way.

7 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS Metric facts. We will say that d is a homogeneous distance on G if it is a continuous distance on G satisfying the following conditions (2.5) d(zx, zy) = d(x, y) and d(δ r (x), δ r (y)) = rd(x, y) x, y, z G, r > 0. Important examples of homogeneous distances are the well known Carnot-Carathéodory distance and those constructed in [14]. It is easily seen that two homogeneous distances are always equivalent. We will denote by B(x, r) and B E (x, r), respectively, the open balls of center x and radius r with respect to a (fixed) homogeneous distance d and the Euclidean distance on R n G. For r > 0 we introduce the boxes Box(0, r) := {y R n : y j < r j j = 1,..., ι} = ( r, r) n 1 ( r 2, r 2 ) n 2 ( r ι, r ι ) nι Box(x, r) := x Box(0, r), x G. By homogeneity, it is easy to observe for any homogeneous distance d there exists C BB = C BB (d) 1 such that (2.6) Box(x, r/c BB ) B(x, r) Box(x, C BB r). We will also use the notation Box µ E (0, r) := {y Rµ : y j < r j = 1,..., µ} = ( r, r) µ. When given 0 < s < r and a linear subspace W of R µ we pose Box µ W (0; r, s) := {y Boxµ E (0, r) : π W (y) < s}, where π W (y) is the canonical projection of y on W. If w 1,..., w H is an orthonormal basis of W we clearly have (2.7) {y Box µ E (0, r) : y, w i < s H i = 1,..., H} Box µ W (0; r, s). From now on, a homogeneous distance d is fixed. We will use several times the following simple fact. Lemma 2.3. There exists C = C(d) > 0 with the following property. For any fixed r 1, x B E (0, r) and j {1,..., ι} there exists x G such that (2.8) (2.9) x 1 = = x j = 0, d(x, x) Cr 1/j x h x h Cr 2 for any h = j + 1,..., ι. Proof. In the case j = 1, we define x = x ( x 1, 0,..., 0) = (x 1,..., x ι ) ( x 1, 0,..., 0) = (0, x 2 + O(r 2 ),..., x ι + O(r 2 )), where the last equality follows from (2.4) and O( ) is understood with respect to the Euclidean norm. By (2.6) we have whence (2.8) follows. d(x, x) = d(0, ( x 1, 0,..., 0)) C BB r

8 8 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE We now argue by induction on j 2, assuming the existence of some x such that x 1 = = x j 1 = 0, d(x, x) Cr 1/(j 1) and x h x h Cr 2 for any h = j,..., ι. Defining x = x (0,..., 0, x j, 0,..., 0), applying both (2.3) and (2.4) we obtain x = ( x 1,..., x ι ) (0,..., 0, x j, 0,..., 0) = (0,..., 0, 0, x j+1 + O(r 2 ),..., x ι + O(r 2 )). Thus, by inductive hypothesis we get x = (0,..., 0, 0, x j+1 + O(r 2 ),..., x ι + O(r 2 )). As a result, we arrive at the following inequalities d(x, x) d(x, x) + d( x, x) Cr 1/(j 1) + d(0, (0,..., 0, x j, 0,..., 0)) that complete the proof. Cr 1/j + C BB x j 1/j = Cr 1/j + C BB x j + O(r 2 ) 1/j Cr 1/j Hausdorff measures and coverings. For the sake of completeness, we recall the definitions of Hausdorff measures. Let q 0 and δ > 0 be fixed; we define { } H q δ (E) := inf (diam E i ) q : E i E i, diam E i < δ i=1 { } S q δ (E) := inf (diam B i ) q : E i B i, B i = B(x i, r i ) balls, diam B i < δ. i=1 The q-dimensional Hausdorff measure of E G is H q (E) := lim δ 0 + Hq δ (E) while the q-dimensional spherical Hausdorff measure of E is The Hausdorff dimension of E is S q (E) := lim δ 0 + Sq δ (E). dim H E := inf{q : H q (E) = 0} = sup{q : H q (E) = + }. It is well-known that dim H G coincides with the homogeneous dimension Q := n 1 + 2n 2 + +ιn ι of G. The standard Euclidean Hausdorff measure on G = R n is denoted by H q. For more information on the properties of these measures, see for instance [11, 24, 27]. We state without proof the following simple fact. Proposition 2.4. Let θ > 0 and let E X, where X is a metric space. If for all ɛ (0, 1) the set E can be covered by N ɛ balls of radius ɛ β with N ɛ C ɛ q and C > 0 independent from ɛ, then the Hausdorff dimension of E is not greater than q/β. The following result, see e.g. [27, Theorem 3.3], will be useful in the sequel.

9 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 9 Theorem 2.5 (5r-covering). Let (X, d) be a separable metric space and E X; let r > 0 be fixed. Then there exists a subset F E at most countable such that E x F B(x, 5r) and B(x, r) B(x, r) = x, x F, x x Multi-indices, degrees and maximal dimension. We denote by I p the set of those multi-indices α = (α 1,..., α p ) {1,..., n} p such that 1 α 1 < < α p n. We also set d(α) := d α1 + + d αp. We denote by D(p) the maximum integer d(α) when α varies in I p. We call this number the maximal dimension, that is uniquely defined for any given p {1, 2,..., n}. Clearly, D(n) equals the homogeneous dimension Q of G. The maximal dimension can be computed in the following way. Define l = l(p) by imposing l := ι if p n ι (2.10) ι ι n j < p n j otherwise. j=l+1 j=l Clearly, l depends on p and it can be equivalently defined by l(p) := d n+1 p, that represents the lowest possible degree among tangent vectors of span{x n p+1,..., X n }, where (X 1,..., X n ) is an adapted basis of the stratified Lie algebra G. It is also easy to see that (2.11) D(p) = ι j=l(p)+1 ( jn j + l(p) p ι j=l(p)+1 n j ), where the two summations in (2.11) have to be understood as 0 when l(p) = ι. We also set ι (2.12) r p := p n j 1, so that j=l(p)+1 (2.13) p = r p + n l(p) n ι and D(p) = l(p)r p + (l(p) + 1)n l(p) ιn ι. It is worth noticing that D(p) = β + (p), where β + is the upper dimension comparison function for G, introduced in [4] Degree of submanifolds, projections and subdilations. By Σ G, we denote a p-dimensional Lipschitz submanifold of G. We define the singular set Σ := {x Σ : C x Σ is not a p-dimensional subspace of R n }, where C x Σ is the (Euclidean) tangent cone to Σ at x, i.e., { } C x Σ := tv R n x : t 0, v = lim i x i x i for some sequence (x x i ) i N Σ with x i x. We have the following fact.

10 10 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE Proposition 2.6. For any p-dimensional Lipschitz submanifold Σ G, we have (2.14) H D(p) (Σ ) = 0. Proof. Since H p (Σ ) = 0, by [4, Proposition 3.1] our claim immediately follows. Given a point x Σ \ Σ, we denote by τ Σ (x) its tangent vector, i.e., the p- dimensional multivector associated with the p-plane C x Σ. We can write τ Σ (x) = α I p c α X α, where X α := X α1... X αp. We then define the degree d Σ (x) of Σ at x as d Σ (x) := max{d(α) : α I p and c α 0}. The degree of Σ is d(σ) := max{d Σ (x) : x Σ \ Σ }. Clearly, we have d(σ) D(p). The underlying metric g on G gives a natural scalar product on multivectors, whose norm will be denoted by. If g is any Riemannian metric on G, then at any x G we have a canonically defined scalar product on any Λ p S x where S x T x G is a p-dimensional subspace. We will denote by g,x its corresponding norm. Definition 2.7. Let g be any Riemannian metric on G, let Σ be a p-dimensional Lipschitz submanifold and let x Σ \ Σ. We define the unit tangent p-vector with respect to g as follows (2.15) τ Σ, g (x) = τ Σ(x) τ Σ (x) g,x, where τ Σ (x) is any tangent p-vector of Σ at x. Dilations of G canonically extend to dilations on multivectors as follows for all v 1,..., v p G, therefore we have (Λ p δ r )(v 1... v p ) = (δ r v 1 )... (δ r v p ) (Λ p δ r )(X α ) = (Λ p δ r )(X α1... X αp ) = r dα 1 + +dαp X α = r d(α) X α. Definition 2.8. A p-vector v Λ p G is homogeneous of degree l N\{0} if (Λ p δ r )v = r l v for all r > 0. Our scalar product on Λ p (G) allows us to introduce the following canonical projections, hence homogeneous multivectors of different degrees are orthogonal. Definition 2.9. Let p, D N be such that 1 p D D(p). Let us introduce the linear subspace Λ D,p (G) of Λ p G made by all homogeneous p-multivectors of degree D. With respect to the scalar product of G, the following orthogonal projection π D : Λ p (G) Λ D,p (G) is uniquely defined. We say that π D is the projection of degree D. If we consider a p-vector t Λ p S x with S x T x G, a pointwise projection π D,x (t) is automatically

11 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 11 defined, taking the left translated multivector (Λ p dl x 1)(t) Λ p (T 0 G), identifying T 0 G with G and applying π D to this translated multivector, hence π D,x (t) = (Λ p dl x ) π D (Λ p dl x 1)(t). To simplify notation, both projection π D applied to v Λ p G and π D,x applied to w Λ p S x will be also denoted by (v) D and (w) D,x, respectively. Remark The previous notions allow us to consider the density function with respect to g, defined as Σ x (τ Σ, g (x)) D. This function naturally appears in the representation of the D-dimensional intrinsic measure of a submanifold. The following lemma will be useful in the sequel. It can be proved repeating exactly the same arguments of [23, Lemma 3.1]. Lemma Let Σ G be a p-dimensional Lipschitz submanifold and fix x Σ\Σ. Then we can find a graded basis X 1,..., X n of G and a basis v 1,..., v p of T x Σ such that, writing v j = n i=1 C ijx i (x), we have (2.16) C := (C ij ) i=1,...,n = j=1,...,p Id α Id α Id αι where α k are integers satisfying 0 α k n k and α α ι = p. The symbols 0 and denote null and arbitrary matrices of the proper size, respectively. We have ι (2.17) d Σ (x) = kα k. Remark As already observed in [23, Remark 3.2], the previous lemma along with its proof are understood to hold also in the case where some α k possibly vanishes. In this case the α k columns of (2.16) containing Id αk and the corresponding vectors vj k are meant to be absent. In a similar way, if α k = n k, then the null sub-matrix is missing from the corresponding diagonal box. The integers α 1,..., α ι of Lemma 2.11 define a sub-grading for a p-dimensional subspace of R n, so that in analogy with the integers m j defined in (2.1), we set (2.18) µ 0 = 0 and µ k = k=1 k α l for all k = 1,..., ι. l=1 This new grading allows us to define for every j {1,..., p} the subdegree σ j defined as follows (2.19) σ j := k if and only if µ k 1 < j µ k for some k {1,..., ι}.

12 12 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE The corresponding subdilations λ r : R p R p are defined as follows λ r (ξ 1,..., ξ p ) = (r σ 1 ξ 1, r σ 2 ξ 2,..., r σp ξ p ) for all r > Transversal points and transversal submanifolds. Let us fix a p-dimensional Lipschitz submanifold Σ and consider x Σ \ Σ, where Σ is its singular set. We say that x is transversal if d Σ (x) equals the maximal dimension D(p). We say that Σ is a transversal submanifold if Σ contains at least one transversal point; clearly, this is equivalent to the condition d(σ) = D(p). Remark For hypersurfaces, transversal points coincide with noncharacteristic points and when p n n 1 a p-dimensional submanifold of G is transversal if and only if it is non-horizontal, according to the terminology of [21]. The following corollary is an easy consequence of the fact that X i (0) = e i. Corollary Under the assumptions and notations of Lemma 2.11, the point x is transversal if and only if the following conditions hold (2.20) α ι = n ι, α ι 1 = n ι 1,..., α l+1 = n l+1, α l = r p, α l 1 = 0,..., α 1 = 0, where l = l(p) is defined by (2.10) and r p is defined in (2.12). If x = 0 is transversal, then the vectors v 1,..., v p in Lemma 2.11 constitute the columns of the matrix Id rp (2.21) C 0 = (Cij) 0 = 0 Id nl Id nι The previous corollary shows that at transversal points the associated grading given by the integers of (2.18) and (2.19) yields j µ 0 = = µ l 1 = 0, µ l = r p and µ l+j = r p + n l+i for all j = 1,..., ι l, therefore the subdegrees are the following ones (2.22) σ 1 = l,..., σ rp = l and σ rp+s+ j i=1 n l+i = σ µ l+j +s = l + j + 1 for all s = 1,..., n l+j+1 and j = 0, 1,..., ι l 1, where the term j i=1 n l+i in the previous formulae is meant to be zero when j = 0. i=1 3. Blow-up at transversal points This section is devoted to the proof of Theorem 1.1 stated in the Introduction. We have first to recall some more notions and fix other auxiliary objects. First of all g will denote any auxiliary Riemannian metric on G. The corresponding Riemannian surface measure induced on a C 1 smooth submanifold Σ G will be denoted by µ.

13 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 13 Definition 3.1. A graded metric g on G is fixed and we set B = {x G : d(0, x) < 1}, where d is a homogeneous distance. The stratified group G is seen as an abstract vector space and S denotes one of its p-dimensional linear subspaces. We consider any simple p-vector τ associated to S. Then we define the metric factor (3.1) θg(τ) d = H p (S B). Here denotes the Euclidean metric on G with respect to a fixed graded basis (X 1,..., X n ) and the sets S and B are represented with respect to the associated coordinates of the first kind. Remark 3.2. In the previous definition, any other simple p-vector λτ with λ 0 defines the same subspace S. Conversely, whenever a simple p-vector ζ is associated to S, that is ζ = ζ 1 ζ p and (ζ 1,..., ζ p ) is basis of S, then ζ = t τ for some t 0. Remark 3.3. The metric factor only depends on the Riemannian metric g and the homogeneous distance d. In fact, under the assumptions of Definition 3.1, let us consider another graded basis (Y 1,..., Y n ) with associated coordinates (y 1,..., y n ) of the first kind. Then the linear change of variables from these coordinates to the original coordinates (x 1,..., x n ) associated to (X 1,..., X n ) is an isometry of R n, hence the number (3.1) is preserved under the coordinates (y i ). About the statement of Theorem 1.1, we wish to clarify that the Lie subgroup Π Σ (x) appearing in (1.3) is also homogeneous in the sense that is closed under dilations. Furthermore, it is a p-dimensional homogeneous subgroup of G of the form Z H l+1 H ι, where Z H l is a linear space of dimension r p. The integers l and r p are defined in (2.10) and (2.12). In particular, Π Σ (x) is also a normal subgroup. The same subgroup is more conveniently defined later in the proof of Theorem 1.1, see (3.25). Proof of Theorem 1.1. First of all, our claim allows us to assume that there exists an open neighbourhood U R p of the origin such that Ψ : U Σ is a C 1 smooth diffeomorphism with Ψ(0) = x. Defining the translated submanifold Σ x = l x 1(Σ), we observe that d Σx (0) = d Σ (x) = D(p) = d(σ) = d(σ x ). We consider the translated diffeomorphism φ = l x 1 Ψ, with φ : U Σ x. Taking into account Corollary 2.14, we have a graded basis of left invariant vector fields X 1,..., X n and linearly independent vectors v 1,..., v p T 0 Σ x such that the matrix C 0 = (Cij), 0 defined by v j = n j=1 C0 ijx i (0) = n j=1 C0 ije i is given by (2.21). The vector fields X i in our coordinates have the form n (3.2) X i = a l i e l and their (nonconstant) coefficients satisfy (see e.g. [15]) { δ (3.3) a l l i = i d l d i homogeneous polynomial of degree d l d i otherwise, l=1

14 14 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE where homogeneity refers to intrinsic dilations of the group. After a linear change of variable on φ, we can also assume that ti φ(0) = v i for all i = 1,..., p, where each v i is the i-th column of (2.21). Let π 0 : R n R p be the projection π 0 (x 1,..., x n ) = (x ml 1 +1,..., x ml 1 +r p, x ml +1,..., x n ). Thus, d(π 0 φ)(0) is invertible and the inverse mapping theorem provides us with new variables y = (y ml 1 +1,..., y ml 1 +r p, y ml +1,..., y n ) such that Σ x is represented near the origin by γ = φ (π 0 φ) 1, which can be written as follows ( γ1 (y),..., γ ml 1 (y), y m l 1+1,..., y ml 1+r p, γ ml 1+r p+1(y),..., γ ml (y), y ml +1,..., y n ) and it is defined in some smaller neighbourhood ( c 1, c 1 ) p U for some c 1 > 0. Furthermore, since d(π 0 φ)(0) is the identity mapping of R p, we get (3.4) ( 1 γ)(0) = v 1, ( 2 γ)(0) = v 2,... ( p γ)(0) = v p, so that we have continuous functions C ij (y) with C ij (0) = Cij 0 such that n (3.5) ( j γ)(y) = C ij (y)x i (γ(y)) for all j = 1,..., p. i=1 Due to the structure of C 0 given in (2.21), whenever σ j = l, or equivalently when j = 1,..., r p, we have (3.6) C ij (y) = δ i ml 1,j + o(1) for any i such that d i l. When l < σ j ι, or equivalently in the case j > r p and j = µ σj 1 + 1,..., µ σj, we get { δi mσj (3.7) C ij (y) = 1,j µ σj + o(1) if d 1 i = σ j or 1 i m σj 1 n σj o(1) if d i > σ j. Let us introduce the C 1 smooth homeomorphism η : R p R p as follows ( t1 σ 1 (3.8) η(t) = sgn(t 1 ),..., t ) p σp sgn(t p ), σ p σ 1 where its inverse mapping is given by the formula ( ) ζ(τ) = sgn(τ 1 ) σ 1 σ1 τ 1,..., sgn(τ p ) σp σ p τ p and all σ j satisfy (2.22). We consider the C 1 smooth reparametrization Γ(t) = γ ( η(t) ) with partial derivatives n (3.9) tj Γ(t) = t j σj 1 ( j γ)(η(t)) = t j σ j 1 C ij (η(t))a s i (Γ(t)) e s for all j = 1,..., p, where we have used both (3.2) and (3.5). We first observe that s,i=1 (3.10) Γ i (t) = o( t d i ) for 1 d i < l. In fact, we have γ(0) = 0 and η(t) = O( t l ), hence { o( t d i ) if d (3.11) Γ i (t) = γ i (η(t)) = O( η(t) ) = i < l O( t l ) if d i = l.

15 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 15 The main point is to prove the following rates of convergence (3.12) { Γi (t) = O( t l ) for m l 1 < i m l 1 + r p Γ i (t) = o( t l ) for m l 1 + r p < i m l. Since the first equation of (3.12) is already contained in (3.11), we have nothing to prove in the case r p = n l (because (3.12) does not have the second case). Thus, we will assume that r p < n l and then prove the second formula of (3.12). First of all, we apply (3.9) and compute the following partial derivatives (3.13) tj Γ i (t) = t j σ j 1 ( j γ i )(η(t)) = t j σ j 1 n C kj (η(t))a i k(γ(t)) for all j = 1,..., p and all i = 1,..., n. If 1 j r p, we rewrite the previous sum as ( t j σ j 1 C kj (η(t))a i k(γ(t)) + ) C kj (η(t))a i k(γ(t)). k:d k <l k:d k l As a consequence, taking into account (3.6), it follows that ( tj Γ i (t) = t j l 1 a i j+m l 1 (Γ(t)) + C kj (η(t))a i k(γ(t)) + k:d k <l = t j l 1 ( a i j+m l 1 (Γ(t)) + o(1) + k:d k <l k=1 k:d k l ) C kj (η(t))a i k(γ(t)). ) o(1) a i k(γ(t)) Since d k < l, we have that a i k is a nonconstant homogenous polynomial. It follows that a i k Γ = o(1) and we get ) (3.14) tj Γ i (t) = t j (a l 1 i j+m l 1 (Γ(t)) + o(1) Since m l 1 + j m l 1 + r p < i m l and d ml 1 +j = d i, formula (3.3) implies that a i j+m l 1 is the null polynomial. It follows that (3.15) tj Γ i (t) = o( t l 1 ) whenever 1 j r p and m l 1 + r p < i m l. If r p < j p, then (3.13) implies that tj Γ i (t) = t j σ j 1 n C kj (η(t))a i k(γ(t)) = t j σj 1 O(1) = O( t σj 1 ). k=1 Since in this case σ j > l, we get in particular that (3.16) tj Γ i (t) = o( t l 1 ) whenever r p < j p and 1 i n. Joining (3.15) with (3.16), it follows that Γ i (t) = o( t l 1 ) for all i = m l 1 + r p + 1,..., m l, that proves the second equation of (3.12).

16 16 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE Now, we write explicitly the form of Γ as the composition γ η. By the previous formulae for γ and η, we get Γ(t) = (Γ 1 (t),..., Γ ml 1 (t), t 1 l l sgn(t 1),..., t r p l sgn(t rp ), Γ ml 1 +r l p+1(t),... (3.17)..., Γ ml (t), t r p+1 l+1 l + 1 sgn(t r p+1),..., t p ι ) sgn(t p ). ι The new parametrization γ of Σ x around the origin yields Φ : ( c 1, c 1 ) p Σ defined as Φ := l x γ, that is our adapted parametrization of Σ around x. Taking r > 0 sufficiently small, we have (3.18) µ(σ B(x, r)) r D(p) ˆ = r D(p) Φ 1 (B(x,r)) ( y1 Φ yp Φ)(y) g dy. where µ is the Riemannian surface measure induced by g on Σ. We perform the change of variable y = λ r t, where λ r is the subdilation of the form (3.19) λ r (t 1,..., t p ) = (r l t 1,..., r l t rp, r l+1 t rp+1,..., r l+1 t rp+n l+1,..., r ι t p ) that yields the formula (3.20) µ(σ B(x, r)) r D(p) = ˆ λ 1/r (Φ 1 (B(x,r))) y1 Φ(λ r t) yp Φ(λ r t) g dt. The point is then to study the behaviour of the set λ 1/r ( Φ 1 (B(x, r)) ) as r 0 +. To do this, we will use the formula (3.17) for Γ and the rates of convergence (3.12), taking into account the change of variables (3.8). Since Φ 1 (B(x, r)) = γ 1 (B(0, r)), it follows that (3.21) λ 1/r ( Φ 1 (B(x, r)) ) = { t R p : δ 1/r ( γ(λr t) ) B }, where B = {z G : d(z, 0) < 1}. We observe that γ(λ r t) = Γ(ζ(λ r t)) = Γ(r ζ(t)), therefore the previous rescaled set can be written as follows (3.22) λ 1/r ( Φ 1 (B(x, r)) ) = { t R p : δ 1/r ( Γ(rζ(t)) ) B }. By (3.17), an element t R p of the previous set is characterized by the property that ( Γ1 (rζ(t)),..., Γ m l 1 (rζ(t)), t r r l 1 1,..., t rp, Γ m l 1+r p+1(rζ(t)),... (3.23) r l..., Γ m l (rζ(t)) ), t rp+1,..., t r l p belongs to B. This is a simple consequence of the equalities η(rζ(t)) = λ r η(ζ(t)) = λ r t. We now use both (3.11) and (3.12) to conclude that the element represented in (3.23) converges to ) (3.24) (0,..., 0, t 1,..., t rp, 0,..., 0, t rp+1,..., t p

17 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 17 as r 0 +, uniformly with respect to t that varies in a bounded set. By standard facts on Hausdorff convergence, the previous limit implies the convergence in (1.3) where (3.25) Π Σ (x) := {z G : z 1 = = z ml 1 = z ml 1 +r p+1 = = z ml = 0}. It can be easily seen that Π Σ (x) is the homogeneous subgroup of G associated with the Lie subalgebra span {X ml 1 +1, X ml 1 +2,..., X ml 1 +r p, X ml +1,... X n }. Moreover, the convergence of the element represented in (3.23) to (3.24) also gives (3.26) lim r 0 + µ(σ B(x, r)) r D(p) = H p (B S) ( y 1 Φ yp Φ)(0) g,x where S = {(0,..., 0, t 1,..., t rp, 0,..., 0, t rp+1,..., t p ) R n : t 1,..., t p R} and the metric unit ball B is represented with respect to the same coordinates. Taking into account (3.4) and the matrix (2.21), we have the projection π D(p),0 ( ( y1 γ yp γ)(0) ) = ( X ml 1 +1 X ml 1 +r p X ml +1 X n ) (0) and the formulae yj Φ(x) = dl x ( yj γ(0) ) yield π D(p),x ( ( y1 Φ yp Φ)(0) ) = ( X ml 1 +1 X ml 1 +r p X ml +1 X n ) (x). We have the unit tangent p-vector τ Σ, g (x) = ( y 1 Φ yp Φ)(0) ( y1 Φ yp Φ)(0) g,x then the previous equations for projections give ( τ Σ, g (x) ) 1 D(p),x =. ( y1 Φ yp Φ)(0) g,x As a result, in view of Definition 3.1, the limit (3.26) proves our last claim (1.4). 4. Negligibility of lower degree points in transversal submanifolds The aim of this section is to prove Theorem 1.2 for a C 1 p-dimensional transversal submanifold Σ G, where we define (4.1) Σ c := {x Σ : d Σ (x) < D(p)}. Since Σ is transversal, the subset Σ c plays the role of a generalized characteristic set of Σ. Since any left translation is a diffeomorphism, for each point x Σ there holds (4.2) T 0 (x 1 Σ) = dl x 1(T x Σ). Clearly, a basis for T 0 (x 1 Σ) is given by dl x 1(v 1 ),..., dl x 1(v p ), where the vectors v 1,..., v p are given by Lemma If v j = i C ijx i (x), by the left invariance of X i, we have n (4.3) dl x 1(v j ) = C ij X i (0) for any j = 1,..., p. i=1

18 18 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE In particular, d x 1 Σ(0) = d Σ (x) and 0 (x 1 Σ) c if and only if x Σ c. Taking into account (2.20), we observe that Σ c can be written as the disjoint union (4.4) Σ c = Σ A c Σ B c, where Σ A c and Σ B c are the (possibly empty) sets defined by (4.5) Σ A c := {x Σ c : j l + 1 such that α j < n j } Σ B c := {x Σ c : α j = n j j l + 1 and α l < r p }. The integer l, depending on p, is introduced in (2.10) and the nonnegative integers α 1,..., α ι are defined in Lemma In particular, l will be used throughout this section. We notice that in the case l = 1, we must have α l = r p, hence Σ B c =. We begin by making the further assumption that Σ is of class C 1 and such that Σ φ([0, 1] p ) for some C 1 -regular map φ : [0, 1] p G. By the uniform differentiability of φ, the boundedness of Σ and the continuity of left translations, the following statement holds: for any ɛ > 0, there exists r ɛ > 0 such that (4.6) y, w ɛr r (0, r ɛ ), x Σ, y (x 1 Σ) B E (0, r), w (T 0 (x 1 Σ)), w = 1, where, denotes the Euclidean scalar product. The orthogonal space (T 0 (x 1 Σ)) is understood with respect to the same product. Notice that such coordinates are associated with the basis X 1,..., X n given by Lemma 2.11; in particular, they depend on the chosen basepoint x Σ. The proof of the negligibility stated in Theorem 1.2 stems from the following key lemmata. The proofs of these lemmata could be rather simplified; however, we present them in a form which will be helpful for some refinement provided in Section 5. In both statements, the number r ɛ is as in (4.6). Lemma 4.1. Let Σ be a C 1 submanifold such that Σ φ([0, 1] p ) for some C 1 map φ : [0, 1] p G; let θ := 1/l. Then, there exists a constant C A = C A (Σ) > 0 such that the following property holds. For any x, ɛ, r satisfying (4.7) x Σ A c, ɛ (0, 1) and 0 < r min{ r ɛ, ɛ l }, the set (x 1 Σ) B E (0, r) can be covered by a family {B i : i I} of CC balls with radius r θ such that #I C A ɛ r p θd(p). Proof. From now on, the numbers C i, with i = 1, 2,..., will denote positive constants depending only on Σ, p, G and the fixed homogeneous distance d. For the reader s convenience, we divide the proof into several steps. Step 1. By Theorem 2.5, we get a countable family {B(x i, r θ ) : i I} such that (4.8) x i (x 1 Σ) B E (0, r) (x 1 Σ) B E (0, r) i I B(x i, r θ ) B(x i, r θ /5) B(x h, r θ /5) = when i h.

19 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 19 We have to estimate #I. By Lemma 2.3, for any i I there exists x i such that (4.9) x 1 i = = x l i = 0, d(x i, x i ) Cr 1/l = Cr θ, x h i x h i Cr 2 for any h = l + 1,..., ι. Therefore, taking into account (2.6), we achieve (4.10) B(x i, r θ ) B( x i, (1 + C)r θ ) Box( x i, C 1 r θ ). Let us also point out that both (4.9) and the fact that x i B E (0, r) give (4.11) x i C 2 r. Step 2. Let us prove that there exists C 3 > 0 such that, for any i I, there holds (4.12) Box( x i, C 1 r θ ) Ω, where we have set Ω := ( C 3 r θ, C 3 r θ ) n 1 ( C 3 r 2θ, C 3 r 2θ ) n 2 ( C 3 r lθ, C 3 r lθ ) n l Box µ E (0, C 3r) and µ := n m l. To this aim we fix y Box(0, C 1 r θ ), that is (4.13) y j < (C 1 r θ ) j j = 1,..., ι, and prove that x i y Ω. By explicit computation (4.14) x i y = (0,..., 0, x l+1 i,..., x ι i) (y 1,..., y ι ) where we have used = (y 1,..., y l, x l+1 i + y l+1, x l+2 i + y l+2 + O(r 1+θ ),..., x ι i + y ι + O(r 1+θ )) (2.3) for the coordinates in the layers 1,..., l + 1; (2.4) for the coordinates in the layers l + 2,..., ι, together with (4.11) and the fact that y = O(r θ ). Here and in the sequel, all the quantities O( ) are uniform. From (4.14) and (4.13) it follows immediately that x i y Ω, and (4.12) follows. Step 3. We have not used the fact that x Σ A c yet. By definition, there exists j l + 1 such that α j < n j. We can also assume that j is maximum, i.e., that α j = n j for any j > j; set ν := n j + n j n ι = n j + α j α ι. The last ν rows of the matrix C given by Lemma 2.11 constitute a ν p matrix M of the form 0 0 Id α j Id α j M = Id n j+1 0 = Id n j+1 + +n ι Id nι Since M has only α j +n j+1 + +n ι < ν nonzero columns, there exists a vector z R ν such that z = 1 and z is orthogonal to any of the columns of M. Therefore, the vector w := (0, z) R n R n ν R ν is orthogonal to any of the columns of C. By (4.2) and

20 20 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE (4.3), taking into account that X k (0) = xk, these columns generate T 0 (x 1 Σ). As a result, since j > l, we are lead to the validity of the following conditions w (T 0 (x 1 Σ)) (4.15) w = 1. w 1 = w 2 = = w ml = 0 Step 4. To refine the inclusion (4.12), we will use the properties (4.15). By (4.6) one has x i, w ɛr for any i I. Define w := (w ml +1, w ml +2,..., w n ) R µ, where µ = n m l > ν is the same number of Step 2. By (4.14) and (4.15), for any y Box(0, C 1 r θ ) we have x i y, w = (y 1,..., y l, x l+1 i + y l+1, x l+2 i + y l+2 + O(r 1+θ ),..., x ι i + y ι + O(r 1+θ )), (0,..., 0, w l+1,..., w ι ) ( x l+1 i,..., x ι i), w + (y l+1,..., y ι ), w + O(r 1+θ ) = (x l+1 i,..., x ι i), w + O(r (l+1)θ ) + O(r 1+θ ) ɛr + O(r 1+θ ), where the second equality is justified by (4.9) and (4.13) and the last inequality follows from (l + 1)θ = 1 + θ. Since all the previous O( )s are uniform with respect to the index i, we get ( x i y) µ, w ɛr + C 4 r 1+θ (1 + C 4 )ɛr, where ( x i y) µ is the vector made by the last µ coordinates of ( x i y) µ and we used the fact that, by (4.7), r θ = r 1/l ɛ. Thus, by (2.7) and (4.12) we obtain that Box( x i, C 1 r θ ) Ω, where we have set Ω := ( C 3 r θ, C 3 r θ ) n 1 ( C 3 r 2θ, C 3 r 2θ ) n 2 ( C 3 r lθ, C 3 r lθ ) n l Box µ span w (0; C 3 r, C 5 ɛr). As a consequence, by (4.10) we get B(x i, r θ /5) Box( x i, C 1 r θ ) Ω for all i I. Step 5. We are ready to estimate #I. The volume of Ω is equal to a = C 6 ɛ r θ(n 1+2n 2 +ln l )+µ = C 6 ɛ r θ(n 1+2n 2 +ln l )+n l+1 + +n ι, while each B(x i, r θ /5) has volume b = C 7 r θ(n 1+2n 2 + +ιn ι). Taking into account that the CC balls B(x i, r θ /5) are pairwise disjoint and contained in Ω, we have #I a b = C 6 C 7 ɛ r n l+1+ +n ι θ((l+1)n l+1 + +ιn ι) (2.13) = C 6 C 7 ɛ r p rp θ(d(p) lrp) = C 6 C 7 ɛ r p θd(p)+(θl 1)rp which proves the claim and concludes the proof of the lemma. While more subtle at certain points, the proof of Lemma 4.2 follows the same lines of the previous one. For the reader s benefit, we will try to make the analogies between the two proofs as evident as possible. We recall again that r ɛ is as in (4.6).

21 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 21 Lemma 4.2. Under the assumptions of Lemma 4.1 and l 2, there exists C B = C B (Σ) > 0 such that the following property holds. For any x, ɛ, θ, r satisfying (4.16) x Σ B c, ɛ (0, 1), 1 l < θ 1 l 1 and 0 < r min{ r ɛ, ɛ 1/(lθ 1) }, the set (x 1 Σ) B E (0, r) can be covered by a family {B i : i I} of CC balls with radius r θ such that #I C B ɛ H r p θd(p) (lθ 1)(n l r p), where H = H(x) := n l α l and the integers α j = α j (x) are those given by Lemma Proof. We follow the same convention of Lemma 4.1 about the constants C i. Step 1. By the 5r-covering theorem we can cover (x 1 Σ) B E (0, r) by a family of CC balls {B(x i, r θ ) : i I} such that (4.8) holds. We have once more to estimate #I. By Lemma 2.3, for any i I there exists x i such that (4.17) x 1 i = = x l 1 i = 0, d(x i, x i ) Cr 1/(l 1) Cr θ and x h i x h i Cr 2 for any h = l,..., ι. Therefore B(x i, r θ /5) B(x i, r θ ) B( x i, (1 + C)r θ ) Box( x i, C 8 r θ ). Again (4.18) x i C 2 r. Step 2. Let us prove that there exists C 9 > 0 such that, for any i I, there holds (4.19) Box( x i, C 8 r θ ) Ω, where now (4.20) Ω := ( C 9 r θ, C 9 r θ ) n 1 ( C 9 r 2θ, C 9 r 2θ ) n 2 ( C 9 r (l 1)θ, C 9 r (l 1)θ ) n l 1 Box µ E (0, C 9r) and µ := n m l 1 = n l + + n ι. As before we fix y Box(0, C 8 r θ ), (4.21) y j < (C 8 r θ ) j j = 1,..., ι and prove that x i y Ω. Reasoning as in Step 2 in the proof of Lemma 4.1 we get (4.22) x i y = (0,..., 0, x l i,..., x ι i) (y 1,..., y ι ) = (y 1,..., y l 1, x l i + y l, x l+1 i + y l+1 + O(r 1+θ ),..., x ι i + y ι + O(r 1+θ )) where we have used (2.3), (2.4), (4.18) and the fact that y = O(r θ ). All the quantities O( ) are uniform. The inclusion (4.19) follows from (4.18), (4.22) and the fact that y j < (C 8 r θ ) j = C j 8r jθ C j 8r j/l C j 8r j = l,..., ι. Step 3. Since x Σ B c we have by definition α l < r p and α j = n j j l + 1.

22 22 VALENTINO MAGNANI, JEREMY T. TYSON, AND DAVIDE VITTONE Therefore the last µ rows of the matrix C from Lemma 2.11 constitute a µ p matrix M of the form 0 0 Id αl Id αl M = Id nl+1 0 = Id nl+1 + +n ι Id nι There are α l + n l n ι nonzero columns of M; therefore, the columns of M span a vector subspace of R µ of dimension at most α l + n l n ι. Since µ (α l + n l n ι ) = n l α l = H, it follows that there exist H linearly independent vectors z 1,..., z H R µ such that z k = 1 and z k is orthogonal to any of the columns of M for any k = 1,..., H. In particular, the unit vectors w k := (0, z k ) R n R n µ R µ, k = 1,..., H are orthogonal to any of the columns of C, which form a basis of T 0 (x 1 Σ). Setting W := span(w 1,..., w H ) we have W T 0 (x 1 Σ) and dim W = H 1 ; moreover, any vector w W is of the form (4.23) w = (0,..., 0, w l,..., w ι ) = (0, w ) R m l 1 R µ. Step 4. Again we want to refine the inclusion (4.19). By (4.6) there holds x i, w ɛr i I, w W with w = 1. Recalling (4.22) and writing w = (0, w ) R m l 1 R µ as in (4.23), for any y Box(0, C 8 r θ ) we have x i y, w = (y 1,..., y l 1, x l i + y l, x l+1 i + y l+1 + O(r 1+θ ),..., x ι i + y ι + O(r 1+θ )), (0,..., 0, w l,..., w ι ) ( x l i,..., x ι i), w + (y l,..., y ι ), w + O(r 1+θ ) = (x l i,..., x ι i), w + O(r lθ ) + O(r 1+θ ) ɛr + O(r lθ ) + O(r 1+θ ) w W, w = 1 where we used (4.17) and (4.21). Since lθ = (l 1)θ + θ 1 + θ, we have r 1+θ r lθ and thus, since all the O( )s are uniform, x i y, w ɛr + C 10 r lθ max{ɛ, C 10 r lθ 1 }r C 11 ɛ r w W, w = 1, the last inequality following from (4.16). Using (2.7) we can then refine (4.20) to obtain B(x i, r θ /5) Box( x i, C 8 r θ ) Ω i I

23 TRANSVERSAL SUBMANIFOLDS IN STRATIFIED GROUPS 23 where Ω := ( C 9 r θ, C 9 r θ ) n 1 ( C 9 r 2θ, C 9 r 2θ ) n 2 ( C 9 r (l 1)θ, C 9 r (l 1)θ ) n l 1 Box µ W (0; C 9r, C 11 ɛr). Step 5. We can now estimate #I. Since dim W = H, the volume of Ω is a = C 12 ɛ H r θ(n 1+2n 2 +(l 1)n l 1 )+µ = C 12 ɛ H r θ(n 1+2n 2 +(l 1)n l 1 )+n l + +n ι, while each ball B(x i, r θ /5) has volume b = C 7 r θ(n 1+2n 2 + +ιn ι). B(x i, r θ /5) are pairwise disjoint and contained in Ω, we have as claimed. #I a b = C 12 C 7 ɛ H r n l+ +n ι θ(ln l + +ιn ι) (2.13) = C 12 C 7 ɛ H r n l+p r p θ[l(n l r p)+lr p+(l+1)n l+1 + +ιn ι] (2.13) = C 12 C 7 ɛ H r p+(n l r p) θ[l(n l r p)+d(p)] = C 12 C 7 ɛ H r p θd(p) (lθ 1)(n l r p), Since the CC balls Lemma 4.3. Let Σ be a C 1 submanifold such that Σ φ([0, 1] p ) for a C 1 map φ : [0, 1] p G. Then H D(p) (Σ c ) = 0. Proof. Clearly, it will be enough to show that (4.24) H D(p) (Σ A c ) = 0 and H D(p) (Σ B c ) = 0. Step 1. We start by proving the first equality in (4.24); let us follow the same convention of Lemmata 4.1 and 4.2 about the constants C i. Let ɛ (0, 1) and r (0, min{ r ɛ, ɛ l }] be fixed. Since (x, y) x 1 y is locally Lipschitz and φ is Lipschitz, both with respect to the Euclidean distance, we obtain C 13 > 0 such that (4.25) if z 1, z 2 [0, 1] p and z 1 z 2 C 13 r, then φ(z 1 ) 1 φ(z 2 ) < r. Let us divide [0, 1] p, in a standard fashion, into a family of closed subcubes of diameter not greater than C 13 r; in this way there will be less than C 14 r p such subcubes. Let (Q j ) j J be the family of those subcubes with the property that φ(q j ) Σ A c and fix x j φ(q j ) Σ A c. By (4.25) we have x 1 j φ(q j ) (x 1 j Σ) B E (0, r). Writing θ := 1/l, Lemma 4.1 ensures that x 1 j φ(q j ) can be covered by (at most) C A ɛr p θd(p) balls of radius r θ ; by left invariance, the same holds for φ(q j ). In particular, since Σ A c j J φ(q j ) and #J C 14 r p, we have that, for any r (0, min{ r ɛ, ɛ l }], the set Σ A c can be covered by a family of CC balls with radius r θ of cardinality controlled by C A C 14 ɛr θd(p). Therefore H D(p) 2r 1/l (Σ A c ) C A C 14 ɛr θd(p) (2r θ ) D(p) = 2 D(p) C A C 14 ɛ