The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = {r[mid ]k[mid ]eikϕ}k∈ℤ is never a basis of b1, 1 and b∞, 0 for any weighted L1-norm and L∞-norm over D. The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted L1-norms and L∞-norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L1-norms and L∞-norms over [Copf ] where Ω is indeed a basis of b1, 1 and b∞, 0. Moreover, using similar methods, a weight is constructed on D where b∞, ∞ is not isomorphic to l∞ which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.

In this paper we construct new obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group. We also determine the structure of the cokernel of the Johnson homomorphism for degrees 2 and 3.

Some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic $n$-space are studied as an application of the theory of Legendrian singularities.

The statistical properties of endomorphisms under the assumption that the associated Perron–Frobenius operator is quasicompact are considered. In particular, the central limit theorem, weak invariance principle and law of the iterated logarithm for sufficiently regular observations are examined. The approach clarifies the role of the usual assumptions of ergodicity, weak mixing, and exactness.

Sufficient conditions are given for quasicompactness of the Perron–Frobenius operator to lift to the corresponding equivariant operator on a compact group extension of the base. This leads to statistical limit theorems for equivariant observations on compact group extensions.

Examples considered include compact group extensions of piecewise uniformly expanding maps (for example Lasota–Yorke maps), and subshifts of finite type, as well as systems that are nonuniformly expanding or nonuniformly hyperbolic.

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. $A$ is defined over a subfield $F_0$ if there exists an $F_0$-algebra $A_0$ such that $A = A_0 \bigotimes_{F_0} F$. The following are shown. (i) In many cases $A$ can be defined over a rational extension of $k$. (ii) If $A$ has odd degree $n \geq 5$, then $A$ is defined over a field $F_0$ of transcendence degree $\leq {\frac{1}{2}}(n-1)(n-2)$ over $k$. (iii) If $A$ is a $\mathbb{Z}/m \times \mathbb{Z}/2$-crossed product for some $m \geq 2$ (and in particular, if $A$ is any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of two symbol algebras. Consequently, ${\rm M}_m(A)$ can be defined over a field $F_0$ such that ${\rm trdeg}_k(F_0) \leq 4$. (iv) If $A$ has degree 4 then the trace form of $A$ can be defined over a field $F_0$ of transcendence degree $\leq 4$. (In (i), (iii) and (iv) it is assumed that the center of $A$ contains certain roots of unity.)

It is shown that if a locally compact group acts isometrically on a Banach space X leaving a closed subspace M invariant, and if the induced actions on M and X/M are strongly continuous, then the action on X is strongly continuous. Since this may be of interest for one-parameter semigroups, similar results are proved for actions of suitable topological semigroups. Other generalizations are given for (suitable) non-isometric actions, non-locally compact groups, and non-Banach spaces; corollaries concerning 1-cocycles and uniformly continuous actions are given.

Let [hfr ] be a reductive subalgebra of a semisimple Lie algebra [gfr ] and C[hfr ] ∈ U([hfr ]) be the Casimir element determined by the restriction of the Killing form on [gfr ] to [hfr ]. The paper studies eigenvalues of C[hfr ] on the isotropy representation [mfr ]≃[gfr ]/[hfr ]. Some general estimates connecting the eigenvalues and the Dynkin indices of [mfr ] are given. If [hfr ] is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C[hfr ] on exterior powers of [mfr ] is connected with possible dimensions of commutative Lie subalgebras in [mfr ], thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C[hfr ] on ∧ [mfr ]. More generally, a similar picture arises if [hfr ] = [gfr ]Θ, where Θ is an automorphism of finite order m and [mfr ] is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.

A finite non-trivial group G is called a Hurwitz group if it is an image of the infinite triangle group

formula here

Thus G is a Hurwitz group if and only if it can be generated by an involution and an element of order 3 whose product has order 7. The history of Hurwitz groups dates back to 1879, when Klein [9] was studying the quartic

formula here

of genus 3. The automorphism group of this curve has order 168 = 84(3−1), and it is isomorphic to the simple group PSL2(7), which is generated by the projective images of the matrices

formula here

with product

formula here

and so is a Hurwitz group. In 1893, Hurwitz [7] proved that the automorphism group of an algebraic curve of genus g (or, equivalently, of a compact Riemann surface of genus g) always has order at most 84(g−1), and that, moreover, a finite group of order 84(g−1) can act faithfully on a curve of genus g if and only if it is an image of Δ(2, 3, 7).

The problem of determining which finite simple groups are Hurwitz groups has received considerable attention. In [10], Macbeath classified the Hurwitz groups of type PSL2(q); there are infinitely many of them. In [1] Cohen proved that no group PSL3(q) is a Hurwitz group except PSL3(2), which is isomorphic to PSL2(7). Certain exceptional groups of Lie type, and some of the sporadic groups, are known to be Hurwitz groups. For discussions of the results on these groups we refer the reader to [3, 5, 11].

A previous conjecture is verified for any normal surface singularity which admits a good ${\mathbb C}^*$-action. This result connects the Seiberg–Witten invariant of the link (associated with a certain ‘canonical’ spin$^c$ structure) with the geometric genus of the singularity, provided that the link is a rational homology sphere.

As an application, a topological interpretation is found of the generalized Batyrev stringy invariant (in the sense of Veys) associated with such a singularity.

The result is partly based on the computation of the Reidemeister–Turaev sign-refined torsion and the Seiberg–Witten invariant (associated with any spin$^c$ structure) of a Seifert 3-manifold with negative orbifold Euler number and genus zero.

To a reduced plane curve singularity with r branches there is associated its semigroup, a subsemigroup of ℤr[ges ]0. For an irreducible curve singularity, the description of its semigroup in terms of its set of generators is well known. For a reducible curve there exists a combinatorial description of the semigroup. The semigroup of a plane curve singularity with several branches is described in terms of the minimal set of generators constructed in a natural geometric way.

Throughout this paper k denotes a fixed commutative ground ring. A Cohen–Macaulay complex is a finite simplicial complex satisfying a certain homological vanishing condition. These complexes have been the subject of much research; introductions can be found in, for example, Björner, Garsia and Stanley [6] or Budach, Graw, Meinel and Waack [7]. It is known (see, for example, Cibils [8], Gerstenhaber and Schack [10]) that there is a strong connection between the (co)homology of an arbitrary simplicial complex and that of its incidence algebra. We show how the Cohen–Macaulay property fits into this picture, establishing the following characterization.

A pure finite simplicial complex is Cohen–Macaulay over k if and only if the incidence algebra over k of its augmented face poset, graded in the obvious way by chain lengths, is a Koszul ring.

We revise the concept of compact tripotent in the bidual space of a JB*-triple. This concept was introduced by Edwards and Rüttimann generalizing the ideas developed by Akemann for compact projections in the bidual of a C*-algebra. We also obtain some characterizations of weak compactness in the dual space of a JC*-triple, showing that a bounded subset in the dual space of a JC*-triple is relatively weakly compact if and only if its restriction to any abelian maximal subtriple $C$ is relatively weakly compact in the dual of $C$. This generalizes a very useful result by Pfitzner in the setting of C*-algebras. As a consequence we obtain a Dieudonné theorem for JC*-triples which generalizes the one obtained by Brooks, Saitô and Wright for C*-algebras.

The classification theorem of compact surfaces states that every topological orientable compact surface is homeomorphic to a sphere or to a ‘torus’ of genus $g$, with $g\,=\,1,2,\dots$. It is proved in the paper that every hyperbolic Riemann surface except for $\bold D\setminus\{0\}$ can be decomposed into basic pieces of only a few different types: Y-pieces, funnels and half-disks. As a corollary of this result, the generalization of the classification theorem to non-compact surfaces is obtained.

We prove that every limit group acts geometrically on a CAT(0) space with the isolated flats property.

Let G be a connected semisimple group over an algebraically closed field K of characteristic p>0, and [gfr ]=Lie (G). Fix a linear function χ∈[gfr ]* and let [zfr ][gfr ](χ) denote the stabilizer of χ in [gfr ]. Set [Nscr ]p([gfr ]) ={x∈[gfr ][mid ]x[p]=0}. Let [Cscr ]χ([gfr ]) denote the category of finite-dimensional [gfr ]-modules with p-character χ. In [7], Friedlander and Parshall attached to each M∈Ob([Cscr ]χ([gfr ])) a Zariski closed, conical subset [Vscr ][gfr ](M)⊂[Nscr ]p([gfr ]) called the support variety of M. Suppose that G is simply connected and p is not special for G, that is, p≠2 if G has a component of type Bn, Cn or F4, and p≠3 if G has a component of type G2. It is proved in this paper that, for any nonzero M∈Ob([Cscr ]χ([gfr ])), the support variety [Vscr ][gfr ](M) is contained in [Nscr ]p([gfr ])∩[zfr ][gfr ](χ). This allows one to simplify the proof of the Kac–Weisfeiler conjecture given in [18].

In this paper we study the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra of an operator are subsets in the complex plane which describe where the resolvent is large in norm. The study of such subsets allows us to understand the stability of the spectrum under perturbations and the possible calculation of ‘false eigenvalues’ far from the spectrum by algorithms for computing eigenvalues. The rotated harmonic oscillator is the simplest classic non-self-adjoint quadratic Hamiltonian, which has already been studied by Davies and Boulton. In one of his works, Boulton states a conjecture about the pseudo-spectra of this operator, which describes the instabilities for high energies. We can deduce this conjecture from a study of Dencker, Sjöstrand and Zworski, which gives bounds on the resolvent for a semi-classical pseudo-differential operator in a very general setting. In the present paper, we give a more elementary proof of this result using only some non-trivial localization scheme in the frequency variable.

The paper gives a formula for the probability that a randomly chosen elliptic curve over a finite field has a prime number of points. Two heuristic arguments in support of the formula are given as well as experimental evidence. The paper also gives a formula for the probability that a randomly chosen elliptic curve over a finite field has kq points where k is a small number and q is a prime.

Let $Y$ be a reduced irreducible projective curve of arithmetic genus $g\ge 2$ with at most ordinary nodes as singularities. For a subsheaf $F$ of rank $r'$, degree $d'$ of a torsion-free sheaf $E$ of rank $r$, degree $d$, let $s(E,F) = r'd-rd'$. Define $s_{r'}(E) = {\rm min} s(E,F)$, where the minimum is taken over all subsheaves of $E$ of rank $r'$. For a fixed $r', s_{r'}$ defines a stratification of the moduli space $U(r,d)$ of stable torsion-free sheaves of rank $r$, degree $d$ by locally closed subsets $U_{r',s}$. We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank $r'$ for $s\le r'(r-r')(g-1)$ (respectively $s<r'(r-r')(g-1)$). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve $Y$ is nonspecial.

(Almost strongly minimal) generalized n-gons are constructed for all n [ges ] 3 for which the automorphism group acts transitively on the set of ordered ordinary (n + 1)-gons contained in it, a new class of BN-pairs thus being obtained. Through the construction being modified slightly, 2ℵ0 many non-isomorphic almost strongly minimal generalized n-gons are obtained for all n [ges ] 3, none of which interprets an infinite group. Furthermore, a characterization is given of all graphs whose simple cycles all have length 2n for some n [ges ] 3.

In the 1960s, Richard J. Thompson introduced a triple of groups F ⊆ T ⊆ G which, among them, supplied the first examples of infinite, finitely presented, simple groups [14] (see [6] for published details), a technique for constructing an elementary example of a finitely presented group with an unsolvable word problem [12], the universal obstruction to a problem in homotopy theory [8], and the first examples of torsion free groups of type FP∞ and not of type FP [5]. In abstract measure theory, it has been suggested by Geoghegan (see [3] or [9, Question 13]) that F might be a counterexample to the conjecture that any finitely presented group with no non-cyclic free subgroup is amenable (admits a bounded, non-trivial, finitely additive measure on all subsets that is invariant under left multiplication). Recently, F has arisen in the theory of groups of diagrams over semigroup presentations [10], and as the object of questions in the algebra of string rewriting systems [7]. For more extensive bibliographies and more results on Thompson's groups and their generalizations see [1, 4, 6].

A persistent peculiarity of Thompson's groups is their ability to pop up in diverse areas of mathematics. This suggests that there might be something very natural about Thompson's groups. We support this idea by showing (Theorem 1.1 below) that PLo(I), the group of piecewise linear (finitely many changes of slope), orientation-preserving, self-homeomorphisms of the unit interval, is riddled with copies of F: a very weak criterion implies that a subgroup of PLo(I) must contain an isomorphic copy of F.