Boundary structure of hyperbolic -manifolds admitting annular fillings at large distance

We show that if a hyperbolic -manifold with a union of tori admits two annular Dehn fillings at distance , then is bounded by at most three tori.

     

The length of a shortest closed geodesic and the area of a -dimensional sphere

Let be a Riemannian manifold homeomorphic to . The purpose of this paper is to establish the new inequality for the length of a shortest closed geodesic, , in terms of the area of . This result improves previously known inequalities by C.B. Croke (1988), by A. Nabutovsky and the author (2002) and by S. Sabourau (2004).

     

Simultaneous non-vanishing of twists

Let be a newform of even weight , level and character and let be a newform of even weight , level and character . We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet -functions, in the context of twisted modular -functions associated to and . Using this result, we find a lower bound in terms of for the number of primitive Dirichlet characters modulo prime whose twisted product -functions are non-vanishing at a fixed point with .

     

Universal localization of triangular matrix rings

If is a triangular matrix ring, the columns and are f.g. projective -modules. We describe the universal localization of which makes invertible an -module morphism , generalizing a theorem of A. Schofield. We also describe the universal localization of -modules.

     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

Asymptotically harmonic spaces in dimension 3

Let be a Hadamard manifold of dimension whose sectional curvature satisfies and whose curvature tensor satisfies for suitable constants and . We show that is of constant sectional curvature provided is asymptotically harmonic. This was previously only known if admits a compact quotient.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

Hodge structures on posets

Let be a poset with unique minimal and maximal elements and . For each , let be the vector space spanned by -chains from to in . We define the notion of a Hodge structure on which consists of a local action of on , for each , such that the boundary map intertwines the actions of and according to a certain condition.

We show that if has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of into Hodge pieces.

We consider the case where is , the poset of subsets of with cardinality divisible by is fixed, and is a multiple of . We prove a remarkable formula which relates the characters of acting on the Hodge pieces of the homologies of the to the characters of acting on the homologies of the posets of partitions with every block size divisible by .

     

An ascending HNN extension of a free group inside

We give an example of a subgroup of which is a strictly ascending HNN extension of a non-abelian finitely generated free group . In particular, we exhibit a free group in of rank which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005). The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold which is a surface bundle over the circle. In particular, most of comes from the fundamental group of a surface fiber. A key feature of is that there is an element of in with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group we construct is actually free.

     

On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

A strong hot spot theorem

A real number is said to be -normal if every -long string of digits appears in the base- expansion of with limiting frequency . We prove that is -normal if and only if it possesses no base- ``hot spot'. In other words, is -normal if and only if there is no real number such that smaller and smaller neighborhoods of are visited by the successive shifts of the base- expansion of with larger and larger frequencies, relative to the lengths of these neighborhoods.

     

The rank of elliptic curves with rational 2-torsion points over large fields

Let be a number field, an algebraic closure of , the absolute Galois group , the maximal abelian extension of and an elliptic curve defined over . In this paper, we prove that if all 2-torsion points of are -rational, then for each , has infinite rank, and hence has infinite rank.

     

The Banach algebra generated by a contraction

Let be a contraction on a Banach space and the Banach algebra generated by . Let be the unitary spectrum (i.e., the intersection of with the unit circle) of . We prove the following theorem of Katznelson-Tzafriri type: If is at most countable, then the Gelfand transform of vanishes on if and only if

     

Global dominated splittings and the Newhouse phenomenon

We prove that given a compact -dimensional boundaryless manifold , , there exists a residual subset of the space of diffeomorphisms such that given any chain-transitive set of , then either admits a dominated splitting or else is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes given by Bonatti, Diaz, and Pujals (2003).

It follows from the above result that given a -generic diffeomorphism , then either the nonwandering set may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else exhibits infinitely many periodic sinks/sources (the `` Newhouse phenomenon"). This result answers a question of Bonatti, Diaz, and Pujals and generalizes the generic dichotomy for surface diffeomorphisms given by Mañé (1982).

     

The weak Dirichlet problem for Baire functions

Let be a simplex and a compact subset of the set of all extreme points of . We show that any bounded function of Baire class on can be extended to a function of affine class on . Moreover, can be chosen in such a way that .

     

A short proof of Bing's characterization of

We give a short proof of Bing's characterization of : a compact, connected 3-manifold is if and only if every knot in is isotopic into a ball.

     

Spaces that admit hypercyclic operators with hypercyclic adjoints

A continuous linear operator is hypercyclic if there is an such that the orbit is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces does contain such an operator. We prove that for any infinite-dimensional Banach space with a shrinking symmetric basis, such as and any , there is an operator , where both and are hypercyclic.

     

The spherical Paley-Wiener theorem on the complex Grassmann manifolds SU

We prove the Paley-Wiener theorem for the spherical transform on the complex Grassmann manifolds SUSU   U. This theorem characterizes the -biinvariant smooth functions on the group that are supported in the -invariant ball of radius , with less than the injectivity radius of , in terms of holomorphic extendability, exponential growth, and Weyl invariance properties of the spherical Fourier transforms , originally defined on the discrete set of highest restricted spherical weights.

     

Isomorphic -subspaces in Orlicz-Lorentz sequence spaces

Given a decreasing weight and an Orlicz function satisfying the -condition at zero, we show that the Orlicz-Lorentz sequence space contains an -isomorphic copy of , if and only if the Orlicz sequence space does, that is, if , where and are the Matuszewska-Orlicz lower and upper indices of , respectively. If does not satisfy the -condition, then a similar result holds true for order continuous subspaces and of and , respectively.