Proceedings of the London Mathematical Society - recent issues

Principal non-commutative torus bundles

Echterhoff, S., Nest, R., Oyono-Oyono, H. — 2009-06-11

In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally RKK-trivial. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to Tⁿ-equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal Tⁿ-bundles with H-flux, as studied by Mathai and Rosenberg which possess ‘classical’ T-duals.

On the role of convexity in functional and isoperimetric inequalities

Milman, E. — 2009-06-11

This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but in general this is not an equivalence. We show that under certain convexity assumptions (for example, for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz–Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz’ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, q-log-Sobolev inequalities (q [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ) curvature–dimension condition of Bakry–Émery.

Transversality in families of mappings

Wall, C. T. C. — 2009-06-11

If N and P are smooth manifolds, and Q is a smooth submanifold of P, the set of maps of N to P transverse to Q is residual. However given a family F of smooth mappings, we cannot expect to be able to deform the family to make each member of the family transverse to Q. The development of generic transversality conditions runs in parallel with that of stability theory. We develop a convenient notation, and establish equivalence of different transversality conditions. The first main result expresses equivalence between F inducing a versal unfolding of any multi-germ, multi-transversality of F to contact orbits relative to projection on the parameter space U, and local stability of the projection to U of the pre-image of Q. For global stability we need Q to be closed, the deformation F to be proper, and Q or N compact.

Corresponding results also hold for topological stability in the Thom–Mather sense, and an appropriate notion of topological versality.

To obtain results with Q a stratified subset of P requires an extensive study of theories of contact equivalence relative to a subset of the target. Here we need an analyticity hypothesis on Q; for deeper study of finite determinacy, we need a holonomic condition; even so, the theory is much more complicated. In general, there are properties corresponding to the former ones, but many are weaker. Our second main result, a direct analogue of the first, is only obtained under a hypotheses that F has property (G) at all points and is transverse to Q.

In the topological case, while some results can be obtained, the failure of direct analogues to some basic tools is a bar to further progress.

A class of noncommutative projective surfaces

Rogalski, D., Stafford, J. T. — 2009-06-11

Let A = _i≥0A_i be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A) = k(X)[t, t^–1; ], where is an automorphism of the integral projective surface X. Then we prove that A can be written as a naïve blowup algebra of a projective surface X birational to X. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in qgr-A will always be in (1-1) correspondence with the closed points of the scheme X.

Representations of Lie superalgebras in prime characteristic I

Wang, W., Zhao, L. — 2009-06-11

We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p > 2. A superalgebra generalization of the celebrated Kac–Weisfeiler conjecture is formulated, which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We establish the conjecture for basic classical Lie superalgebras.

On pseudo-harmonic maps in conformal geometry

Kokarev, G. — 2009-06-11

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We discuss existence theorems and obtain an extension of Siu's strong rigidity to Kähler–Weyl geometry. Other applications include topological obstructions to the existence of Kähler–Weyl structures. For example, we show that no co-compact lattice in SO(1, n), n > 2, can be the fundamental group of a compact Kähler–Weyl manifold of certain type.

Constructing smooth manifolds of loop spaces

Stacey, A. — 2009-06-11

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite-dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions for the model space which, if satisfied, mean that a smooth structure exists. We also show how various desired properties can be derived from the model space; for example, topological properties such as paracompactness. We pay particular attention to the fact that the loop spaces that can be defined in this way are all homotopy equivalent; and also to the action of the circle by rigid rotations.

Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

Sela, Z. — 2009-06-11

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion-free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first-order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion-free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion-free hyperbolic groups are elementarily equivalent.

A quantitative version of the Besicovitch projection theorem via multiscale analysis

Tao, T. — 2009-04-20

By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but weak) upper bound on the average projection of the nth generation of a product Cantor set.

On the logarithmic comparison theorem for integrable logarithmic connections

Calderon Moreno, F. J., Narvaez Macarro, L. — 2009-04-20

Let X be a complex analytic manifold, D X a free divisor with jacobian ideal of linear type (for example, a locally quasi-homogeneous free divisor), j: U = X – D X the corresponding open inclusion, an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of on U. In this paper we prove that the canonical morphisms are isomorphisms in the derived category of sheaves of complex vector spaces for k >> 0 (locally on X).

On the cohomology algebra of some classes of geometrically formal manifolds

Grosjean, J.-F., Nagy, P.-A. — 2009-04-20

We investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal Kähler metric are parallel with respect to the Levi–Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is shown to be flat. Finally we prove that a 6-dimensional manifold with b₁ != 1, b₂ ≥ 2 and not having the real cohomology algebra of T³ x S³ carries a symplectic structure as soon as it admits a formal metric.

Structure and finiteness properties of subdirect products of groups

Bridson, M. R., Miller, C. F. — 2009-04-20

We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of three free or surface groups and to a solution of the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show that there is no algorithm to determine the first homology of a finitely generated subgroup.

Topological stable rank of nest algebras

Davidson, K. R., Ji, Y. Q. — 2009-04-20

We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr (T(N)) = 2 occurs only when N is of ordinal type less than ² and the dimensions of the atoms grows sufficiently quickly. We introduce general results on ‘partial matrix algebras’ over a Banach algebra. This is used to obtain an inequality akin to Rieffel's formula for matrix algebras over a Banach algebra. This is used to give further insight into the nest case.

Schur-Weyl duality for orthogonal groups

Doty, S., Hu, J. — 2009-04-20

We prove Schur–Weyl duality between the Brauer algebra B_n(m) and the orthogonal group O_m(K) over an arbitrary infinite field K of odd characteristic. If m is even, then we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of n-tensor space Vⁿ in the Brauer algebra B_n(m) is also given.

{delta}-sequences and evaluation codes defined by plane valuations at infinity

Galindo, C., Monserrat, F. — 2009-04-20

We introduce the concept of -sequence. A -sequence generates a well-ordered semigroup S in Z² or R. We explain how to construct (and to compute parameters of) the dual code of any evaluation code associated with a weight function defined by from the polynomial ring in two indeterminates to a semigroup S as above. We prove that this is a simple procedure that can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of -sequences of the different existing types, and so this paper helps know and use a new, large set of codes.

Small gaps between products of two primes

Goldston, D. A., Graham, S. W., Pintz, J., Yildirim, C. Y. — 2009-04-20

Let q_n denote the nth number that is a product of exactly two distinct primes. We prove that q_n+1 – q_n ≤ 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if is any positive integer, then (q_n+ – q_n) ≤ e^– (1+o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms.

Generalised Euler characteristics of Selmer groups

Zerbes, S. L. — 2009-04-20

Let E be an elliptic curve defined over a number field F, and let p ≥ 5 be a prime. In this paper, we study the structure of the Selmer group of E, over p-adic Lie extensions F/F. In particular, under certain global and local conditions on F we relate the generalised Gal(F/F)-Euler characteristic of Sel(E/F) to the generalised Euler characteristic of the Selmer group over the cyclotomic Z_p-extension of F. This invariant generalises the classical Euler characteristic to the case when rank_ZE(F) > 0. Moreover, we show that the global and local conditions on F are satisfied for a large class of p-adic Lie extensions of F.

Quivers with potentials associated to triangulated surfaces

Labardini-Fragoso, D. — 2009-04-20

We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin–Shapiro–Thurston, and quivers with potentials (QPs) and their mutations introduced by Derksen–Weyman–Zelevinsky. To each ideal triangulation of a bordered surface with marked points, we associate a QP, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective QPs are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the QPs associated to its triangulations are rigid and hence non-degenerate.

New lower bounds on subgroup growth and homology growth

Lackenby, M. — 2009-02-26

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subgroup growth of any finitely generated group. This is achieved by free non-abelian groups and is slightly faster than exponential. As a consequence, we obtain good estimates on the number of covering spaces of a hyperbolic 3-manifold with given covering degree. We also obtain slightly weaker information on the number of covering spaces of closed 4-manifolds with non-positive Euler characteristic. The results on subgroup growth follow from a new theorem, which places lower bounds on the rank of the first homology (with mod p coefficients) of certain subgroups of a group. This is proved using a topological argument.

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

Basu, S., Kettner, M. — 2009-02-26

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in R, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in . More precisely, we prove the following. Let R be a real closed field and let P = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) ≤ 2, deg_X(P_i) ≤ d, 1 ≤ i ≤ m. Let S R^+k be a semi-algebraic set, defined by a Boolean formula without negations, with atoms of the form P ≥ 0, P ≤ 0, P P. Let : R^+k -> R^k be the projection on the last k coordinates. Then the number of stable homotopy types amongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).

Angled decompositions of arborescent link complements

Futer, D., Gueritaud, F. — 2009-02-26

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements.

New bounds for Szemeredi's theorem, I: progressions of length 4 in finite field geometries

Green, B., Tao, T. — 2009-02-26

Let k ≥ 3 be an integer, and let G be a finite abelian group with |G| = N, where (N, (k – 1)!) = 1. We write r_k(G) for the largest cardinality |A| of a set A G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that r_k(Z/NZ) = o_k(N). It is known, in fact, that the estimate r_k(G) = o_k(N) holds for all G. There have been many papers concerning the issue of finding quantitative bounds for r_k(G). A result of Bourgain states thatr₃(G) << N(log log N/log N)^1/2 for all G. In this paper we obtain a similar bound for r₄(G) in the particular case G = Fⁿ, where F is a fixed finite field with char (F) != 2, 3 (for example, F = F₅). We prove that r₄(G) << _F N(log N)^–c for some absolute constant c > 0. In future papers we will treat the general abelian groups G, eventually obtaining a comparable result for an arbitrary G.

Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality

Tolsa, X. — 2009-02-26

In this paper we study some questions in connection with uniform rectifiability and the L² boundedness of Calderón–Zygmund operators (CZOs). We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients that are related to the Jones’ β numbers. We also use these new coefficients to prove that n-dimensional CZOs with odd kernel of type C² are bounded in L²(µ), if µ is an n-dimensional uniformly rectifiable measure.

The Riemann mapping theorem for semianalytic domains and o-minimality

Kaiser, T. — 2009-02-26

We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result, we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of .

Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

Daniel, B., Hauswirth, L. — 2009-02-26

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil ₃ endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family of annuli is used to prove a vertical half-space theorem which is then applied to prove that each complete minimal graph in Nil ₃ is entire. Also, it is shown that the sister surface of an entire minimal graph in Nil ₃ is an entire constant mean curvature (CMC) 1/2 graph in H² x R, and vice versa. This gives a classification of all entire CMC 1/2 graphs in H² x R. Finally we construct properly embedded CMC 1/2 annuli in H² x R.

Roots in the mapping class groups

Bonatti, C., Paris, L. — 2009-02-26

The purpose of this paper is to study the roots in the mapping class groups. Let be a compact oriented surface, possibly with boundary, let P be a finite set of punctures in the interior of , and let M (, P) denote the mapping class group (relative to the boundary) of (, P). We prove that if is of genus 1 and has nonempty boundary, then each f M () has at most one m-root up to conjugation for all m ≥ 1. We prove that, however, if is of genus at least 2, then there exist f, g M (, P) such that f² = g², f is not conjugate to g, and none of the conjugates of f commutes with g. Afterwards, we focus our study on the roots of the pseudo-Anosov elements. We prove that if != , then each pseudo-Anosov element f M(, P) has at most one m-root for all m ≥ 1, but if = then there exist two pseudo-Anosov elements f, g M () (explicitly constructed) such that f^m = g^m for some m ≥ 2, f is not conjugate to g, and none of the conjugates of f commutes with g. Finally, we show that if is a pure subgroup of M (, P) and f , then f has at most one m-root in for all m ≥ 1. Note that there are finite-index pure subgroups in M(, P).

Powers of sequences and recurrence

Frantzikinakis, N., Lesigne, E., Wierdl, M. — 2009-02-26

We study recurrence and multiple recurrence properties along the kth powers of a given set of integers. We show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mendès-France, which links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher-order recurrence and make a related conjecture.

Small-time versions of Strassen's law for Levy processes

Maller, R. A. — 2009-02-26

We study aspects of the ‘small-time’ behaviour (as t 0) of a Lévy process X(t), obtaining a very general small-time version of Strassen's almost sure (a.s.) functional law of the iterated logarithm (LIL) for random walks. The class of Lévy processes for which this holds is characterised by an explicit analytic condition on the Lévy measure of X, related to an analogous condition of Kesten for a generalised (large-time) random walk LIL. Both centred and uncentred versions of the small-time result are proved. Subsidiary results concerning functional weak convergence of X(t) to Brownian motion as t 0 are shown to be equivalent to the main a.s. results. The quadratic variation process of X is considered, and applications via continuous functionals are suggested.