Volumes des représentations sur un corps local

Let be a locally compact second countable group, F a local field of characteristic zero and G an F-almost-simple F-algebraic group. In this paper we study the space X(,G) of Zariski-dense representations : G = G(F) using the natural morphism of cohomological functors * : H*(G, ·) H*(, ·) (where H denotes the continuous cohomology).First let F be a p-adic field. We completely describe the relations between the geometry and the cohomology of G : using geometric properties of the Bruhat-Tits building of G we construct natural cocycles for any irreducible cohomological representation of G. We then adapt these results to the case where the field F is archimedean.Using these cocycles we obtain a simple cohomological characterization of representations with bounded image.Our main result is then the construction, using the previous cocycles and dynamical properties at infinity of , of cohomological invariants (called volumes) on the space X(,G). These volumes describe how the image () goes to infinity in G. They have coefficients in the natural universal infinite-dimensional representation L(, )$\mathbb{C}$ of .In the case where is a cocompact lattice of SO(n, 1) or SU(n, 1), we use these volumes to produce new non-trivial numerical invariants on X(,G), which refine previously known invariants.

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Volumes des représentations sur un corps local

     

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

A family of multiply extended grids

We construct an infinite family{ _n}_n=5 of finite connected graphs _n that are multiple extensions of the well-known extended grid discovered in [1] (which is isomorphic to ₅). The graphs _n are locally _n–1 forn > 5, and have the following property: the automorphism groupG(n) of _n permutes transitively the maximal cliques of _n (which aren-cliques) and the stabilizer of somen-clique of _n inG(n) induces _n on the vertices of. Furthermore we show that the clique complexes of the graphs _n are simply connected.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Über ebene affine Inzidenzgruppen

For a translation plane P with respect to f we consider the group of collineations generated by all elations fixing f and a point F of f. All subgroups or are determined which operate regularly on the points of the affine plane P. Group-theoretic and operating properties of the groups are stated especially for the finite and the Desarguesian cases. In the latter case the companion NL-near modules are constructed. Finally we characterize the groups within PGL(3, K) with commutative field K of finite characteristic.     

Abelianness of Mumford–Tate Groups Associated to Some Unitary Groups

In this paper, we investigate the action of the -cohomology of the compact dual of a compact Shimura Variety S() on the -cohomology of S()> under a cup product. We use this to split the cohomology of S() into a direct sum of (not necessarily irreducible) -Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on S() are Abelian. As another application, we show that all classes of Hodge type (1,1) in H² of unitary four-folds associated to the group U(2,2) are algebraic.     

Mortality of iterated Gallai graphs

For a finite or infinite graphG, theGallai graph (G) ofG is defined as the graph whose vertex set is the edge setE(G) ofG; two distinct edges ofG are adjacent in (G) if they are incident but do not span a triangle inG. For any positive integert, thetth iterated Gallai graph ^t (G) ofG is defined by ( ^t–1(G)), where ⁰(G):=G. A graph is said to beGallai-mortal if some of its iterated Gallai graphs finally equals the empty graph. In this paper we characterize Gallai-mortal graphs in several ways.     

Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

In this work we consider a system of partial differential operators D ₁,D ₂ on K=[0,+[×R, whose eigenfunctions are the functions (x,t), (x,t)K, =((R0)×N)(0×[0,+[), which are related to the Laguerre functions for ((R 0)×N)(0,0) and which are the Bessel functions for (0×[0,+[). We provide K and with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on .     

Boundary control of parabolic systems: Regularity of optimal solutions

Boundary control problems for the linear, parabolic equations and a quadratic performance index are considered. The controls are allowed to be in the spaceL ²[OT;L²()], where is a boundary. Exploiting the semigroup approach, it is shown that optimal control belongs toL ²[OT;H^1/2()] and, as a consequence, optimal trajectory belongs toL ¹[OT;H¹()]. This result is obtained for two kinds of domains. The first are the domains withC -boundary and the second are the domains being the parallelepipeds.     

Groups acting simply transitively on the vertices of a building of typeà 2, I

If a group acts simply transitively on the vertices of an affine building with connected diagram, then must be of typeà _n–1 for somen2, and must have a presentation of a simple type. The casen=2, when is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when is the building ofG=PGL(3,K),K a local field, and when G. We exhibit such a group whenK=F _q ((X)),q any prime power. Our study leads to combinatorial objects which we calltriangle presentations. These triangle presentations give rise to some new buildings of typeà ₂.     

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Compact Clifford–Klein Forms of Homogeneous Spaces of SO(2, n)

A homogeneous space G/H is said to have a compact Clifford–Klein form if there exists a discrete subgroup of G that acts properly discontinuously on G/H, such that the quotient space \G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2, n), such that G/H has a compact Clifford–Klein form, but our classification is not quite complete when n is odd. The work reveals new examples of homogeneous spaces of SO(2, n) that have compact Clifford–Klein forms, if n is even. Furthermore, we show that if H is a closed, connected subgroup of G = SL(3, R), and neither H nor G/H is compact, then G/H does not have a compact Clifford–Klein form, and we also study noncompact Clifford–Klein forms of finite volume.     

The Proalgebraic Completion of Rigid Groups

A finitely generated group is called representation rigid (briefly, rigid) if for every n, has only finitely many classes of simple representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.     

On the dimensions of automorphism groups of 8-dimensional ternary fields I

Let be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed subgroup of its automorphism group which is taken with the compact-open topology. It is proved that is either isomorphic to the compact exceptional Lie group G₂, or the (covering) dimension of is at most 11. This bound can be decreased to 10, if the ternary fixed fieldF of is connected.     

Fuchsian groups,finite simple groups and representation varieties

Let be a Fuchsian group of genus at least 2 (at least 3 if is non-oriented). We study the spaces of homomorphisms from to finite simple groups G, and derive a number of applications concerning random generation and representation varieties. Precise asymptotic estimates for |Hom(,G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G|^()+1+o(1), where () is the measure of . We then prove that a randomly chosen homomorphism from to G is surjective with probability tending to 1 as |G|. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties , where is GL_n(K) or a simple algebraic group over K, an algebraically closed field of arbitrary characteristic. A key ingredient of our approach is character theory, involving the study of the zeta function ^G(s)=(1)^-s, where the sum is over all irreducible complex characters of G.     

On Locally Projective Graphs of Girth 5

Let be a graph and G be a 2-arc transitive automorphism group of . For a vertex x let G(x)^(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set (x) of vertices adjacent to x in . Then is said to be a locally projective graph of type (n,q) if G(x)^(x) contains PSL_n(q) as a normal subgroup in its natural doubly transitive action. Suppose that is a locally projective graph of type (n,q), for some n 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on (x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n=4, q=2, has 506 vertices and , and contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2²⁰ vertices.