4,264 research outputs found
Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations
The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has
the structure of a Poisson algebra. Assume \g is complex semi-simple. Then
results of Fomenko- Mischenko (translation of invariants) and A.Tarasev
construct a polynomial subalgebra \cal H = \bf C[q_1,...,q_b] of S(\g) which is
maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of
\g. Let G be the adjoint group of \g and let \ell = rank \g. Identify \g with
its dual so that any G-orbit O in \g has the structure (KKS) of a symplectic
manifold and S(\g) can be identified with the affine algebra of \g. An element
x \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly
independent. Then the set \g^{sreg} of all strongly regular elements is Zariski
open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} is the
set of all regular elements in \g. A Hessenberg variety is the b-dimensional
affine plane in \g, obtained by translating a Borel subalgebra by a suitable
principal nilpotent element. This variety was introduced in [K2]. Defining Hess
to be a particular Hessenberg variety, Tarasev has shown that Hess \subset
\g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then
O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let
O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in
O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R
let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold
of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that
there exists, simultaneously over all O \in R, an explicit polarization (i.e.,
a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg}
simulate, in some sense, the cotangent bundle of Hess(O).Comment: 36 pages, plain te
On abstract commensurators of groups
We prove that the abstract commensurator of a nonabelian free group, an
infinite surface group, or more generally of a group that splits appropriately
over a cyclic subgroup, is not finitely generated.
This applies in particular to all torsion-free word-hyperbolic groups with
infinite outer automorphism group and abelianization of rank at least 2.
We also construct a finitely generated, torsion-free group which can be
mapped onto Z and which has a finitely generated commensurator.Comment: 13 pages, no figur
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
Symmetric spaces of higher rank do not admit differentiable compactifications
Any nonpositively curved symmetric space admits a topological
compactification, namely the Hadamard compactification. For rank one spaces,
this topological compactification can be endowed with a differentiable
structure such that the action of the isometry group is differentiable.
Moreover, the restriction of the action on the boundary leads to a flat model
for some geometry (conformal, CR or quaternionic CR depending of the space).
One can ask whether such a differentiable compactification exists for higher
rank spaces, hopefully leading to some knew geometry to explore. In this paper
we answer negatively.Comment: 13 pages, to appear in Mathematische Annale
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
A predictive phenomenological tool at small Bjorken-x
We present the results from global fits of inclusive DIS experimental data
using the Balitsky-Kovchegov equation with running coupling.Comment: 5 pages, 2 figures, prepared for the Proceedings of 'Hot Quarks 2010
Peripheral separability and cusps of arithmetic hyperbolic orbifolds
For X = R, C, or H it is well known that cusp cross-sections of finite volume
X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds
modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the
(4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a
necessary and sufficient condition for such manifolds to be diffeomorphic to a
cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal
tool in the proof of this classification theorem is a subgroup separability
result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm
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
Overlap properties of geometric expanders
The {\em overlap number} of a finite -uniform hypergraph is
defined as the largest constant such that no matter how we map
the vertices of into , there is a point covered by at least a
-fraction of the simplices induced by the images of its hyperedges.
In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph
expansion for higher dimensional simplicial complexes, it was asked whether or
not there exists a sequence of arbitrarily large
-uniform hypergraphs with bounded degree, for which . Using both random methods and explicit constructions, we answer this
question positively by constructing infinite families of -uniform
hypergraphs with bounded degree such that their overlap numbers are bounded
from below by a positive constant . We also show that, for every ,
the best value of the constant that can be achieved by such a
construction is asymptotically equal to the limit of the overlap numbers of the
complete -uniform hypergraphs with vertices, as
. For the proof of the latter statement, we establish the
following geometric partitioning result of independent interest. For any
and any , there exists satisfying the
following condition. For any , for any point and
for any finite Borel measure on with respect to which
every hyperplane has measure , there is a partition into measurable parts of equal measure such that all but
at most an -fraction of the -tuples
have the property that either all simplices with
one vertex in each contain or none of these simplices contain
Arbitrarily large families of spaces of the same volume
In any connected non-compact semi-simple Lie group without factors locally
isomorphic to SL_2(R), there can be only finitely many lattices (up to
isomorphism) of a given covolume. We show that there exist arbitrarily large
families of pairwise non-isomorphic arithmetic lattices of the same covolume.
We construct these lattices with the help of Bruhat-Tits theory, using Prasad's
volume formula to control their covolumes.Comment: 9 pages. Syntax corrected; one reference adde
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