Simple loop conjecture for limit groups

There are noninjective maps from surface groups to limit groups that don’t kill any simple closed curves. As a corollary, there are noninjective all-loxodromic representations of surface groups to SL(2, ?) that don’t kill any simple closed curves, answering a question of Minsky. There are also examples, for any k, of noninjective all-loxodromic representations of surface groups killing no curves with self-intersection number at most k.     

Transformation formula for exponential sums involving fourier coefficients of modular forms

In 1984 Jutila [5] obtained a transformation formula for certain exponential sums involving the Fourier coefficients of a holomorphic cusp form for the full modular groupSL(2, ?). With the help of the transformation formula he obtained good estimates for the distance between consecutive zeros on the critical line of the Dirichlet series associated with the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper we follow Jutila to obtain a transformation formula for exponential sums involving the Fourier coefficients of either holomorphic cusp forms or certain Maass forms for congruence subgroups ofSL(2, ?) and prove similar estimates for the corresponding Dirichlet series.     

Harmonic analysis of SL(2) over a locally compact field

We carry over the pioneer work of Kunze and Stein concerning representation theory and harmonic analysis on SL(2, R) to the group G = SL(2, K), K a locally compact totally disconnected nondiscrete field. The main result is that convolution by an L^p(G) function, 1 ? p < 2, is a bounded operator on L²(G). To accomplish this result we develop the appropriate estimates (which depend upon the work of Sally et al.) that enable us to apply the Kunze and Stein interpolation theory to the Fourier-Laplace transform for the group G. Best possible estimates are obtained.     

Equivariant Pieri Rule for the homology of the affine Grassmannian

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL _n and a similar formula is conjectured for Sp _2n and SO _2n+1. For SL _n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL _n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.     

Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group <Emphasis Type="Italic">SL</Emphasis>(2)

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for some leftinvariant sub-Riemannian metric on the Lie group SL(2) that is right-invariant relative to the Lie subgroup SO(2) ? SL(2) (in other words, for invariant sub-Riemannian metric on weakly symmetric space (SL(2) × SO(2))/SO(2)).     

The structurally stable linear systems on the half-line are those with exponential dichotomies

Let A be a selft-adjoint operator on the Hilbert space L²Ω, ?) = {u ε L_loc²(Ω)|∫_Ω|² ?(x)dx < + ∞} defined by means of a closed, semibounded, sesquilinear form a(·, ·). We obtain a necessary and sufficuents condition for the spectrum of A to be discrete. We apply this result to a Sturm-Liouville problem for an elliptic operator with discontinuous coefficients defined on an unbounded domain and to the study of the spectrum of a Hamiltonian defined by a pseudodifferential operator.     

The local finiteness of some groups generated by a conjugacy class of order 3 elements

We prove local finiteness for the groups generated by a conjugacy class of order 3 elements whose every pair generates a subgroup that is isomorphic to Z ₃, A ₄, A ₅, SL ₂(3), or SL ₂(5).     

The Existence of Complete Mappings of SL(2, q), q≡1 Modulo 4

In 1955, Hall and Paige conjectured that any finite group with a noncyclic Sylow 2-subgroup admits complete mappings. For the groups GL(2, q), SL(2, q), PSL(2, q), and PGL(2, q) this conjecture has been proved except for SL(2, q), q odd. We prove that SL(2, q), q≡1 modulo 4 admits complete mappings.     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.     

The Groups of Motions of Some Three-Dimensional Maximal Mobility Geometries

We find the groups of motions of eight three-dimensional maximal mobility geometries. These groups are actions of just three Lie groups SL₂(R)ΔN, SL₂(C) _R , and SL₂(R)?SL₂(R) on the space R³, where N is a normal abelian subgroup. We also find explicit expressions for these actions.     

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.     

Affine hypersurfaces admitting a pointwise symmetry

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T _p M) for all p ∈ M, which preserves (pointwise) the affine metric h, the difference tensor K (resp. the cubic form) and the affine shape operator S. In this paper, we deal with locally strongly convex affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K, S) under the action of SO(3) on a Euclidean vector space (V, h) and find a representative (canonical form of K and S) of each SO(3)/G-orbit. Then, we classify hypersurfaces admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart of Z ₂). Besides well-known hypersurfaces (for Z ₂ × Z ₂ we get the locally homogeneous hypersurface (x ₁ ?) 1/2x ₃2 (x ₂ ?) 1/2x ₄2) = 1) we obtain e.g. warped products of two-dimensional affine spheres (resp. quadrics) and curves.     

On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h ? 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL₂(k), SL₃(k), SL₄(k), Sp₄(k), and G ₂, nontrivial isomorphisms of this kind exist for SL₄(k) and G ₂ only. For SL₄(k), there are two simple modules with nontrivial second cohomology and, for G ₂, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.     

Split orders and convex polytopes in buildings

As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of SL₂(Z), Hijikata (1974) [13] defines and characterizes the notion of a split order in M₂(k), where k is a local field. In this paper, we generalize the notion of a split order to Mn(k) for n>2 and give a natural geometric characterization in terms of the affine building for SLn(k). In particular, we show that there is a one-to-one correspondence between split orders in Mn(k) and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the n=2 case in which split orders correspond to geodesics in the tree for SL₂(k) with the split order given as the intersection of the endpoints of the geodesic.     

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SLn(q) in the non-defining characteristic ?, describe restrictions of irreducible modules from GLn(q) to SLn(q), classify complex irreducible characters of SLn(q) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SLn(q).     

A Clifford Bundle Approach to the Differential Geometry of Branes

We first recall using the Clifford bundle formalism (CBF) of differential forms and the theory of extensors acting on \({\mathcal{C}\ell}\) (M, g) (the Clifford bundle of differential forms) the formulation of the intrinsic geometry of a differential manifold M equipped with a metric field g of signature (p, q) and an arbitrary metric compatible connection \({\nabla}\) introducing the torsion (2?1)-extensor field \({\tau}\) , the curvature (2 ? 2) extensor field \({\Re}\) and (once fixing a gauge) the connection (1?2)-extensor \({\omega}\) and the Ricci operator \({\partial \bigwedge \partial}\) (where \({\partial}\) is the Dirac operator acting on sections of \({\mathcal{C}\ell(M, g)}\) ) which plays an important role in this paper. Next, using the CBF we give a thoughtful presentation the Riemann or the Lorentzian geometry of an orientable submanifold M (dim M =  m) living in a manifold M? (such that M? \({\simeq \mathbb{R}^n}\) is equipped with a semi- Riemannian metric g? with signature (p?, q?) and p?+q? = n and its Levi- Civita connection D?) and where there is defined a metric g =  i*g?, where \({i : M \rightarrow}\) M? is the inclusion map. We prove several equivalent forms for the curvature operator \({\Re}\) of M. Moreover we show a very important result, namely that the Ricci operator of M is the (negative) square of the shape operator S of M (object obtained by applying the restriction on M of the Dirac operator ?? of \({\mathcal{C}\ell}\) (M?, g?) to the projection operator P). Also we disclose the relationship between the (1?2)-extensor \({\omega}\) and the shape biform \({\mathcal{S}}\) (an object related to S). The results obtained are used to give a mathematical formulation to Clifford’s theory of matter. It is hoped that our presentation will be useful for differential geometers and theoretical physicists interested, e.g., in string and brane theories and relativity theory by divulging, improving and expanding very important and so far unfortunately largely ignored results appearing in reference [13].     

Periodic magnetic Schrödinger operators: Spectral gaps and tunneling effect

A periodic Schrödinger operator on a noncompact Riemannian manifold M such that H ¹(M, ?) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field, the existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system.