Search

An F -essential subgroup is called a pearl if it is either elementary abelian of order p 2 or non-abelian of order p 3 . In this paper we start the investigation of fusion systems containing pearls: we determine a bound for the order of p-groups containing pearls and we classify the saturated fusion systems on p-groups containing pearls and having sectional rank at most 4.

Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we describe the (intertwining) Hecke algebra $\mathscr{H}(G,K^1)$, that is the convolution $\mathbb{Z}$-algebra of functions from $G$ to $\mathbb{Z}$ that are bi-invariant for $K^1$ and whose supports are a finite union of $K^1$-double cosets. We produce a presentation by generators and relations of this algebra. Finally we prove that the level-$0$ subcategory of the category of smooth representations of $G$ over a unitary commutative ring $R$ such that $p\in R^{\times}$ is equivalent to the category of modules over $\mathscr{H}(G,K^1)\otimes_\mathbb{Z} R$., Comment: 17 pages, minor corrections, updated bibliography

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group., Comment: 11 pages; to appear in Pacific Journal of Mathematics

The intersection growth of a group $G$ is the asymptotic behavior of the index of the intersection of all subgroups of $G$ with index at most $n$, and measures the Hausdorff dimension of $G$ in profinite metrics. We study intersection growth in free groups and special linear groups and relate intersection growth to quantifying residual finiteness., Comment: 20 pages, no figures. Revised version. We extend estimates to polycyclic groups and explain why normal intersection growth is a profinite invariant. Theorem 6.1 previously contained an error which has now been fixed

For a finite cyclic p -group G and a discrete valuation domain R of characteristic 0 with maximal ideal p R the R [ G ] -permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups (cf. Theorem A ). As a consequence any R [ G ] -lattice can be presented by R [ G ] -permutation modules (cf. Theorem C ). The proof of these results is based on a detailed analysis of the category of cohomological G -Mackey functors with values in the category of R -modules. It is shown that this category has global dimension 3 (cf. Theorem E ). A crucial step in the proof of Theorem E is the fact that a gentle R -order category (with parameter p ) has global dimension less than or equal to 2 (cf. Theorem D ).

Let n ≥ 2 and let α ∈ Vnbe an element in the Higman-Thompson group Vn. We study the structure of the centralizer of ∈ Vnthrough a careful analysis of the action of (α) on the Cantor set. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks and flow graphs to assist us in our analysis. A consequence of our structure theorem is that element centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in Vn. © European Mathematical Society.

In this thesis we discuss how, in the context of knot theory, the classifying space of a knot group for the family of meridians arises naturally. We provide an explicit construction of a model for that space, which is particularly nice in the case of a prime knot. We then show that this classifying space controls the behaviour of the finite branched coverings of the knot. We present a 9-term exact sequence for knot groups that strongly resembles the Poitou-Tate exact sequence for algebraic number fields. Finally, we show that the homology of the classifying space behaves towards the former sequence as Shafarevich groups do towards the latter.

We prove that Claas Röver's Thompson-Grigorchuk simple group V G has type F∞. The proof involves constructing two complexes on which V G acts: a simplicial complex analogous to the Stein complex for V , and a polysimplicial complex analogous to the Farley complex for V . We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

W.H.~Mills has determined, for a finitely generated abelian group $G$, the regular subgroups $N \cong G$ of $S(G)$, the group of permutations on the set $G$, which have the same holomorph of $G$, that is, such that $N_{S(G)}(N) = N_{S(G)}(\rho(G))$, where $\rho$ is the (right) regular representation. We give an alternative approach to Mills' result, which relies on a characterization of the regular subgroups of $N_{S(G)}(\rho(G))$ in terms of commutative ring structures on $G$. We are led to solve, for the case of a finitely generated abelian group $G$, the following problem: given an abelian group $(G, +)$, what are the commutative ring structures $(G, +, \cdot)$ such that all automorphism of $G$ as a group are also automorphisms of $G$ as a ring?, Comment: 18 pages. Accepted for publication, Journal of Algebra, March 2017

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated by a couple of examples that might be of independent interest., Comment: 19 pages - minor revisions

We solve the twisted conjugacy problem on Thompson's group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut_+(F) are orbit decidable provided a certain conjecture on Thompson's group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak-Fel'shtyn-Goncalves in [4] showing that F has property R_\infty, and which can be extended to show that Thompson's group T also has property R_\infty., Comment: 31 pages, 2 figures; improved exposition

We consider generalisations of Thompson’s group V, denoted by V r ⁢ ( Σ ) ${V_{r}(\Sigma)}$ , which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, V r ⁢ ( Σ ) ${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type F ∞ ${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type F ∞ ${\operatorname{F}_{\infty}}$ .

It is shown in Lehnert and Schweitzer ('The co-word problem for the Higman-Thompson group is context-free', Bull. London Math. Soc. 39 (2007) 235-241) that R. Thompson's group $V$ is a co-context-free ($co\mathcal {CF}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal {CF}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal {CF}$ group ${\rm QAut}(\mathcal {T}-{2,c})$, which is a group of particular bijections on the vertices of an infinite binary 2-edge-coloured tree, and he conjectures that ${\rm QAut}(\mathcal {T}-{2,c})$ is a universal $co\mathcal {CF}$ group. We show that ${\rm QAut}(\mathcal {T}-{2,c})$ embeds into $V$, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group $V$. In particular, we classify precisely which Baumslag-Solitar groups embed into $V$.

Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin's group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$ of type $\mathrm{FP}$ it is possible to define an Euler-Poincar\'e characteristic $\chi(G)$ which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field $K$ and some other examples.

Let G F be the group of points fixed by the Frobenius endomorphism F of a simple algebraic group G defined in characteristic p . In this paper, we show that if p is a good prime then the size of every U F -conjugacy class, U F a p -Sylow subgroup of G F , is a q -power, where q is the level of F . We also determine the existence of power series inducing isomorphisms between U F and its Lie algebra.

The non-finiteness of the mod-p Schur multiplier of a finitely generated group G with trivial center implies the existence of uncountably many non-residually finite, non-isomorphic central extension groups with kernel C p (see Thm. A). This phenomenon is related to the comparison of the cohomology of the profinite completion G ˆ of a group G and the cohomology of the group G itself (see Thm. B); according to J-P. Serre [8] a group G is good if the cohomologies of G ˆ and G are naturally isomorphic on finite coefficients. It is shown that non-uniform arithmetic lattices of algebraic rank 1 groups over local fields of positive characteristic p are not good (see Thm. C).

In this partly expository paper, we study the set [Formula: see text] of groups of orientation-preserving homeomorphisms of the circle S1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in [Formula: see text]. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T.

A finite group G is called p i -central of height k if every element of order p i of G is contained in the k th -term ζ k (G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω1(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P p is less than or equal to the order of the subgroup Ω1(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N G (P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl p (G) is p-central of height p − 2, then N G (P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]).

In this paper we prove that there exists no function F (m, p) (where the first argument is an integer and the second a prime) such that, if G is a finite permutation p-group with m orbits, each of size at least pF (m,p), then G contains a fixed-point-free element. In particular, this gives an answer to a conjecture of Peter Cameron; see [4], [6].

Maclagan and Smith \cite{MaclaganSmith} developed a multigraded version of Castelnuovo-Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve $C\subseteq \P^a\times\P^b$ $(a, b\geq 2)$ of bidegree $(d_1,d_2)$ with nondegenerate birational projections the ideal sheaf $\mathcal{I}_{C|\P^a\times\P^b}$ is $(d_2-b+1,d_1-a+1)$-regular. We also give an example showing that in some cases this bound is the best possible., Comment: 11 pages

The aim of this paper is studying the groups in which the zeros of every irreducible character are restricted in being p-elements, for some fixed prime p. In particular, we classify the groups whose irreducible characters vanish only on involutions. Some remarkable examples are also presented. © 2008 Elsevier B.V. All rights reserved.

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

This thesis focuses on two problems on l-modular representation theory of p-adic groups. Let F be a non-archimedean local field of residue characteristic p different from l. In the first part, we study block decomposition of the category of smooth modular representations of GL(n; F) and its inner forms.We want to reduce the description of a positive-level block to the description of a 0-level block (of a similar group) seeking equivalences of categories. Using the type theory of Bushnell-Kutzko in the modular case and a theorem of category theory, we reduce the problem to find an isomorphism between two intertwining algebras. The proof of the existence of such an isomorphism is not complete because it relies on a conjecture that we state and we prove for several cases. In the second part we generalize the construction of metaplectic group and Weil representation in the case of representations over un integral domain. We define a central extension of the symplectic group over F by the multiplicative group of an integral domain. We prove that it satisfies the same properties as in the complex case. Cette thèse traite deux problèmes concernant la théorie des représentations l-modulaires d’un groupe p-adique. Soit F un corps local non archimédien de caractéristique résiduelle p différente de l. Dans la première partie, on étudie la décomposition en blocs de la catégorie des représentations lisses `-modulaires de GL(n; F) et de ses formes intérieures. On veut ramener la description d’un bloc de niveau positif à celle d’un bloc de niveau 0 (d’un autre groupe du même type) en cherchant des équivalences de catégories. En utilisant la théorie des types de Bushnell-Kutzko dans le cas modulaire et un théorème de la théorie des catégories, on se ramene à trouver un isomorphisme entre deux algèbres d’entrelacement. La preuve de l’existence d’un tel isomorphisme n’est pas complète car elle repose sur une conjecture qu’on énonce et qui est prouvée pour plusieurs cas. Dans une deuxième partie on généralise la construction du groupe métaplectique et de la représentation de Weil dans le cas des représentations sur un anneau intègre. On construit une extension centrale du groupe symplectique sur F par le groupe multiplicatif d’un anneau intègre et on prouve qu’il satisfait les mêmes propriétés que dans le cas des représentations complexes.

In questa tesi vengono presentati alcuni risultati parziali riguardo la surface group conjecture di Melnikov. La congettura è che ogni gruppo ereditariamente ad un relatore, residualmente finito, non libero e non ciclico sia un surface group. Un surface group è un gruppo isomorfo al gruppo fondamentale di una superficie chiusa. Un surface group ammette una presentazione con una sola relazione ed ogni suo sottogruppo di indice finito è ancora un surface group e quindi un gruppo ad una sola relazione. Chiameremo gruppi ereditariamente ad un relatore quei gruppi ad un relatore con la proprietà che ogni sottogruppo di indice finito sia ancora un gruppo ad un relatore. I surface group hanno dimensione comologica 2 e sono gruppi di dualità. Baumslag ha dimostrato che i surface group sono residualmente finiti. Un primo approccio ad una soluzione del problema è di tipo combinatorico. Alcune proprietà dei gruppi ad un relatore sono determinate da proprietà della relazione. Per esempio, un gruppo ad un relatore è privo di torsione se e solo se la relazione non è una potenza. Un risultato combinatorio più utile e profondo è il Teorema di Identità di Lyndon, che si può dimostrare tramite il calcolo differenziale libero. Se un gruppo ad un relatore è non libero, privo di torsione e liberamente indecomponibile, allora è un gruppo di dualità. Inoltre il Teorema di Identità permette di determinare la struttura del modulo dualizzante. Dopo questi risultati combinatori, si procede ad analizzare i gruppi ereditariamente ad un relatore utilizzando un risultato dovuto a Bieri et al, che hanno dimostrato come i gruppi di Poincaré di dimensione due sono surface groups. Dato un gruppo di dualità ad un relatore G risulta essere un quoziente di ZG e si ha un sollevamento della mappa di augmentazione di ZG ad una mappa dal modulo dualizzante a Z se e solo se la relazione che definisce G è un commutatore. Se il nucleo K di questo sollevamento è banale allora il modulo dualizzante è isomorfo a Z e G è un surface group, per cui K può essere visto come una misura di quanto G sia lontano dall'essere un surface group. Le ipotesi della surface group conjecture hanno notevoli somiglianze ad alcune proprietà dei gruppi di Demushki, che sono gruppi pro-p ad un relatore e gruppi di dualità di Poincaré. La classificazione dei gruppi di Demushkin dovuta a Labue mostra che essi ammettono una presentazione (come pro-p gruppi) simile alla presentazione dei surface group. Si è quindi deciso di esplorare la possibilità di una relazione tra le due situazioni. Dato un gruppo G che soddisfi le ipotesi della surface group conjecture e la cui unica relazione sia un commutatore, si procede a prenderne il completamento pro-p e se ne studia la struttura. Dato che i surface group sono residualmente liberi richiediamo anche che G sia residualmente libero. Dimostriamo quindi che G è p-buono, cioé che il morfismo naturale tra G e il suo completamento induce un isomorfismo tra i rispettivi gruppi di coomologia. Usiamo questa proprietà per caratterizzare il completamento pro-p di G. Teorema. Sia G un gruppo ereditariamente ad un relatore, residualmente finito, non libero e non ciclico, la cui unica relazione sia un commutatore. Allora il completamento pro-p di G è un gruppo di Demushkin orientabile e quindi coincide con il completamento pro-p di un surface group. Utilizzando la classificazione di Labute concludiamo che G deve avere un numero pari di generatori e che la relazione non può trovarsi nel secondo gruppo derivato del gruppo libero il cui quoziente offre la presentazione ad un relatore di G. Inoltre, se G ha solo due generatori abbiamo una risposta positiva alla congettura.

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups., 8 pages, to appear on the Annales math\'ematiques du Qu\'ebec

It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated, $|G:N

The doubly-fed induction generator (DFIG) is a key constituent of energy conversion plants. The control of a DFIG is a challenge, whenever the primary energy supply (e.g., the wind velocity field) is characterised by intermittency. The mathematical model and control of a DFIG rely on the Blondel-Park transformation, which is known to simplify the governing equations. The distinctive feature of this contribution consists of showing how the Blondel-Park transformation derives from a set of conditions to be met by a group. Such a group is shown to exist and to continuously depend on one parameter. The uniqueness of its infinitesimal generator is also shown. As an application, the well-known electric torque theorem is proved in a simple way, which relies on a "product of matrices" formula. The latter, in turn, is a by-product of the axiomatic deduction of the Blondel-Park transformation.

It is shown that for the restricted Zassenhaus algebra \({\mathfrak{W} = \mathfrak{W}(1; n)}\), n > 1, defined over an algebraically closed field \({\mathbb{F}}\) of characteristic 2, any projective indecomposable restricted \({\mathfrak{W}}\) -module has maximal possible dimension \({2^{2^n-1}}\), and thus is isomorphic to some induced module \({{\rm idn}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))}\) for some torus of maximal dimension \({\mathfrak{t}}\). This phenomenon is in contrast to the behavior of finite-dimensional non-solvable restricted Lie algebras in characteristic p > 3 (cf. Feldvoss et al. Restricted Lie algebras with maximal 0-pim, 2014, Theorem 6.3).

The doubly-fed induction generator (DFIG ) is a key constituent of energy conversion plants. The control of a DFIG is a challenge, whenever the primary energy supply (e.g., the wind velocity field) is characterised by intermittency. The mathematical model and control of a DFIG rely on the Blondel-Park transformation, which is known to simplify the governing equations. The distinctive feature of this contribution consists of showing how the Blondel-Park transformation derives from a set of conditions to be met by a group. Such a group is shown to exist and to continuously depend on one parameter. The uniqueness of its infinitesimal generator is also shown. As an application, the well-known electric torque theorem is proved in a simple way, which relies on a “product of matrices” formula. The latter, in turn, is a by-product of the axiomatic deduction of the Blondel-Park transformation.

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam beta-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.

Any morphism $$\phi :\ifmmode\expandafter\hat\else\expandafter\^\fi{G} \to A$$ of profinite groups has maximal l-Frattini quotients $$(\pi ,\beta ),\pi :B \to A,\;\beta :\ifmmode\expandafter\hat\else\expandafter\^\fi{G} \to B,\;{\text{i}}{\text{.e}}{\text{., }}\phi = \pi \circ \beta, \pi$$ is an l-Frattini extension and β is a surjective morphism of profinite groups for which every minimal finite non-trivial l-embedding problem is not weakly solvable. In this paper the case is studied where Ĝ Ĝ is a weakly-orientable l-Poincare duality group of dimension 2 and where A is a finite group whose order is divisible by l. This analysis can be applied for the study of modular towers (Theorem A, Remark 1.2). It is shown that the existence of finite maximal l-Frattini quotients is controlled by an integer r l(A) (Theorem B). In the final section we study properties of the morphism ϕ which imply that for every maximal l-Frattini quotient (π, β), the profinite group B itself is a weakly-orientable l-Poincare duality group of dimension 2 (Theorem C).

The main result of this paper is a graph-theoretic necessary and sufficient condition, for a given set of transvections in SL(n, K) (n > 2 and K a finite field of characteristic not 2 or 3), to generate a group isomorphic to SL (m, L), for some m and some subfield L of K.

Let 1 --> N --> G --> G/N --> 1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n := cd(p)(G) < infinity and if the order of H-k(N, F-p) is finite for k := cd(p)(N), the virtual cohomological p-dimension of G/N equals n - k. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-$p$ case, i.e., one would like to know which pro-$p$ groups occur as maximal pro-$p$ Galois groups, i.e., maximal pro-$p$ quotients of absolute Galois groups. Indeed, pro-$p$ groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group. We define a new class of pro-$p$ groups, called Bloch-Kato pro-$p$ group, whose Galois cohomology satisfies the consequences of the Bloch-Kato conjecture. Also we introduce the notion of cyclotomic orientation for a pro-$p$ group. With this approach, we are able to recover new substantial information about the structure of maximal pro-$p$ Galois groups, and in particular on $\theta$-abelian pro-$p$ groups, which represent the "upper bound" of such groups. Also, we study the restricted Lie algebra and the universal envelope induced by the Zassenhaus filtration of a maximal pro-$p$ Galois group, and their relations with Galois cohomology via Koszul duality. Altogether, this thesis provides a rather new approach to maximal pro-$p$ Galois groups, besides new substantial results., Comment: Ph.D. thesis at Western University (Canada) and Universit\`a di Milano-Bicocca (Italy), 103 pages, 1 figure

Let C be a binary linear code and suppose that its automorphism group contains a non trivial subgroup G. What can we say about C knowing G? In this thesis we collect some answers to this question. We focus on the cases G = C_p, G = C_2p and G = D_2p (p an odd prime), with a particular regard to the case in which C is self-dual. Furthermore we generalize some methods used in other papers on this subject. The third chapter is devoted to the investigation of the automorphism group of a putative self-dual [72; 36; 16] code, whose existence is a long-standing open problem. Last chapter is about semi self-dual codes and new upped bound on their dual distance.