For each finitely presented group $Q$ , a short exact sequence $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ is produced, such that $G$ is residually finite and $N$ is finitely generated.

A Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold is proved, under the condition that $E^* \otimes F$ is Griffiths $k$-positive. This result is applied to find some homotopy groups of the Brill–Noether loci for a generic curve.

Wirtinger and Northcott introduced an inequality bounding the norm of a 2$\Omega$ periodic function by the norm of its derivatives. It is shown here that this type of result occurs for other operators, defined on various domains and spaces.

It is proved that the number of ${\bb F}_q$-rational points of an irreducible projective smooth 3-dimensional geometrically unirational variety defined over the finite field ${\bb F}_q$ with $q$ elements is congruent to 1 modulo $q$. Some Fermat 3-folds, some classes of rationally connected 3-folds and some weighted projective $d$-folds also having this property are given.

This paper introduces baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter $t=0$ (the so-called rational Cherednik algebras at parameter $t=0$), and presents their most basic properties. Baby Verma modules are then used to answer several problems posed by Etingof and Ginzburg, and to give an elementary proof of a theorem of Finkelberg and Ginzburg.

Let $G_0$ and $G$ be the heat kernels of $\Delta$ and $\Delta - V$ respectively. Under some sharp conditions on $V$, a proof is given that $G(x, t; y, 0)/G_0(x, t; y, 0)$ can be bounded from above and below by two positive constants. This largely improves the well-known bounds $C_1 e^{-c_1 |x-y|^2/t} \le G(x, t; y, 0)/G_0(x, t; y, 0) \le C_2 e^{c_2 |x-y|^2/t}$, $t \in (0, T]$. The result answers an open question of V. Liskevich and Yu. Semenov, in the case where $V \ge 0$. A sharp global bound in time is also obtained when $V \ge 0$.

The Hall–Paige conjecture deals with conditions under which a finite group $G$ will possess a complete mapping, or equivalently a Latin square based on the Cayley table of $G$ will possess a transversal. Two necessary conditions are known to be: (i) that the Sylow 2-subgroups of $G$ are trivial or non-cyclic, and (ii) that there is some ordering of the elements of $G$ which yields a trivial product. These two conditions are known to be equivalent, but the first direct, elementary proof that (i) implies (ii) is given here.

It is also shown that the Hall–Paige conjecture implies the existence of a duplex in every group table, thereby proving a special case of Rodney's conjecture that every Latin square contains a duplex. A duplex is a ‘double transversal’, that is, a set of $2n$ entries in a Latin square of order $n$ such that each row, column and symbol is represented exactly twice.

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.

Assuming that the integrated density of states of a Schrödinger operator admits a high-energy asymptotic expansion, the authors give explicit formulae for the coefficients of this expansion in terms of the heat invariants.

This paper provides a unified and elementary proof of four Thompson–Wielandt-like theorems.

A theorem of Beurling and Nevanlinna gives a lower bound for the harmonic measure of a compact set in the unit disk in terms of the harmonic measure of a projection of the set. In this paper, a generalization of that result is proved, involving the sum of harmonic measures evaluated at two points.

Minimal complex surfaces of general type with $p_g = 0$ and $K^2 = 7$ or $8$ whose bicanonical map is not birational are studied. It is shown that if $S$ is such a surface, then the bicanonical map has degree 2 (see Bulletin of the London Mathematical Society 33 (2001) 1–10) and there is a fibration $f : S \rightarrow {\bb P}^1$ such that (i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; (ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$.

Furthermore, if $K^2_S = 8$, then $f$ is an isotrivial fibration with six double fibres, and if $K^2_S = 7$, then $f$ has five double fibres and it has precisely one fibre with reducible support, consisting of two components.

The problem of classifying finitely generated groups up to virtual isomorphism is strictly harder than that of classifying them up to isomorphism, in the sense that there is a Borel reduction from the isomorphism problem to the virtual isomorphism problem, but not vice versa.Research partially supported by NSF Grants.

The paper gives lists of all the primitive permutation groups of squarefree degree. All such groups are either solvable and act on a prime number of points, or are almost simple. Among the almost simple examples, the groups of Lie type have rank at most $2$, or the point stabilizer is a parabolic subgroup.

Let ${\cal D}$ be a complex of non-trivial $p$-subgroups of a finite group $G$, closed under $G$-conjugation. In this paper, a $p$-local geometry $\Delta_p(G; {\cal D})$ is introduced for $G$ associated with the complex ${\cal D}$. The homotopy equivalence between $\Delta_p(G; {\cal D})$ and ${\cal D}$ is also studied.

By means of a simplification in a significant step of Zelmanov's solution of the restricted Burnside problem, the known bounds for the order of finite $m$ -generator groups of exponent $n$ are improved.

A relationship is established between the factorization of $24 n + 1$ and the 5-divisibility of $Q(n)$ , where $Q(n)$ is the number of partitions of $n$ into distinct parts. As an application, an abundance of infinite families of congruences for $Q(n)$ modulo powers of 5 are explicitly exhibited.

If $G$ is a finite group and $V$ is a finite-dimensional ${\rm Q\!\!\!I}\,[G]$-module, $V$ is isomorphic to its contragredient module $V^*$. In general, $V$ need not contain any ${\bb Z}[G]$-lattice which is locally isomorphic to its contragredient lattice. Nevertheless, it turns out that for every $V$ there exists another ${\rm Q\!\!\!I}\,[G]$-module $V^{\prime}$; such that both $V^{\prime}$; and $V \oplus V^{\prime}$; contain ${\bb Z}[G]$-lattices which are locally isomorphic to their contragredient lattices.

This paper presents the rigidity of $\lambda_2$-eigenmaps from $S^{4}$ to itself, and the nonexistence of some $\lambda_2$-eigenmaps, as well as new examples of eigenmaps between spheres.

This paper demonstrates that, in dimensions $n\ge 3$, the metric boundary of a conformal deformation of the unit ball is pathwise connected, and even of bounded turning, provided that the conformal scaling factor satisfies a Harnack inequality and the volume growth of the deformed space is at most euclidean.