4,623 research outputs found
On the bounded cohomology of semi-simple groups, S-arithmetic groups and products
We prove vanishing results for Lie groups and algebraic groups (over any
local field) in bounded cohomology. The main result is a vanishing below twice
the rank for semi-simple groups. Related rigidity results are established for
S-arithmetic groups and groups over global fields. We also establish vanishing
and cohomological rigidity results for products of general locally compact
groups and their lattices
The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
Let k be a global field and let k_v be the completion of k with respect to v,
a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected
algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let
G=\mathbf{G}(k_v). Let \Gamma be an arithmetic lattice in G and let C=C(\Gamma)
be its congruence kernel. Lubotzky has shown that C is infinite, confirming an
earlier conjecture of Serre. Here we provide complete solution of the
congruence subgroup problem for \Gamm$ by determining the structure of C. It is
shown that C is a free profinite product, one of whose factors is
\hat{F}_{\omega}, the free profinite group on countably many generators. The
most surprising conclusion from our results is that the structure of C depends
only on the characteristic of k. The structure of C is already known for a
number of special cases. Perhaps the most important of these is the
(non-uniform) example \Gamma=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the
ring of S-integers in k, with S=\{v\}, which plays a central role in the theory
of Drinfeld modules. The proof makes use of a decomposition theorem of
Lubotzky, arising from the action of \Gamma on the Bruhat-Tits tree associated
with G.Comment: 27 pages, 5 figures, to appear in J. Reine Angew. Mat
Distinction of representations via Bruhat-Tits buildings of p-adic groups
Introductory and pedagogical treatmeant of the article : P. Broussous
"Distinction of the Steinberg representation", with an appendix by Fran\c{c}ois
Court\`es, IMRN 2014, no 11, 3140-3157. To appear in Proceedings of Chaire Jean
Morlet, Dipendra Prasad, Volker Heiermann Ed. 2017. Contains modified and
simplified proofs of loc. cit. This article is written in memory of
Fran\c{c}ois Court\`es who passed away in september 2016.Comment: 33 pages, 4 figure
Conjugacy theorems for loop reductive group schemes and Lie algebras
The conjugacy of split Cartan subalgebras in the finite dimensional simple
case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are
fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie
algebras the affine algebras stand out. This paper deals with the problem of
conjugacy for a class of algebras --extended affine Lie algebras-- that are in
a precise sense higher nullity analogues of the affine algebras. Unlike the
methods used by Peterson-Kac, our approach is entirely cohomological and
geometric. It is deeply rooted on the theory of reductive group schemes
developed by Demazure and Grothendieck, and on the work of J. Tits on buildingsComment: Publi\'e dans Bulletin of Mathematical Sciences 4 (2014), 281-32
A lattice in more than two Kac--Moody groups is arithmetic
Let be an irreducible lattice in a product of n infinite irreducible
complete Kac-Moody groups of simply laced type over finite fields. We show that
if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic
group over a local field and is an arithmetic lattice. This relies on
the following alternative which is satisfied by any irreducible lattice
provided n is at least 2: either is an S-arithmetic (hence linear)
group, or it is not residually finite. In that case, it is even virtually
simple when the ground field is large enough.
More general CAT(0) groups are also considered throughout.Comment: Subsection 2.B was modified and an example was added ther
-Spectral theory of locally symmetric spaces with -rank one
We study the -spectrum of the Laplace-Beltrami operator on certain
complete locally symmetric spaces with finite volume and
arithmetic fundamental group whose universal covering is a
symmetric space of non-compact type. We also show, how the obtained results for
locally symmetric spaces can be generalized to manifolds with cusps of rank
one
A Note on the Equality of Algebraic and Geometric D-Brane Charges in WZW Models
The algebraic definition of charges for symmetry-preserving D-branes in
Wess-Zumino-Witten models is shown to coincide with the geometric definition,
for all simple Lie groups. The charge group for such branes is computed from
the ambiguities inherent in the geometric definition.Comment: 12 pages, fixed typos, added references and a couple of remark
Reduction and approximation in gyrokinetics
The gyrokinetics formulation of plasmas in strong magnetic fields aims at the
elimination of the angle associated with the Larmor rotation of charged
particles around the magnetic field lines. In a perturbative treatment or as a
time-averaging procedure, gyrokinetics is in general an approximation to the
true dynamics. Here we discuss the conditions under which gyrokinetics is
either an approximation or an exact operation in the framework of reduction of
dynamical systems with symmetryComment: 15 pages late
On Motives Associated to Graph Polynomials
The appearance of multiple zeta values in anomalous dimensions and
-functions of renormalizable quantum field theories has given evidence
towards a motivic interpretation of these renormalization group functions. In
this paper we start to hunt the motive, restricting our attention to a subclass
of graphs in four dimensional scalar field theory which give scheme independent
contributions to the above functions.Comment: 54
Recrutement électromyographique des ischio-jambiers selon 30 exercices de rééducation sollicitant le genou
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