We give a linear lower bound on the exponential growth rate of a non-elementary subgroup of a word hyperbolic group, with respect to the number of generators for the subgroup.

In this paper, we give a lower bound of the p-part of the reduced Euler characteristic of the order complex of the centric p-radical subgroups by studying vertices of indecomposable summands of the reduced Lefschetz module. This bound is in fact best possible for at least some groups, and also provides a uniform explanation of the observed phenomenon on the reduced Euler characteristic for some sporadic simple groups.

A group $G$ is said to have the Bergman property (the property of uniformity of finite width) if given any generating $X$ with $X=X^{-1}$ of $G$, we have that $G=X^k$ for some natural $k$, that is, every element of $G$ is a product of at most $k$ elements of $X$. We prove that the automorphism group $\operatorname{Aut}(N)$ of any infinitely generated free nilpotent group $N$ has the Bergman property. Also, we obtain a partial answer to a question posed by Bergman by establishing that the automorphism group of a free group of countably infinite rank is a group of uniformly finite width.

Our first result is a ‘sum-product’ theorem for subsets A of the finite field ${{\mathbb F}_p}$, p prime, providing a lower bound on $\max (|A+A|, |A\cdot A|)$. The second and main result provides new bounds on exponential sums

\[\sum_{x_1,\dots,x_k\in A} \exp(2\pi ix_1\dotsc x_k\xi/p),\]

where $A\subset{{\mathbb F}_p}$.

We investigate the finer fractal structure of the set of points escaping to infinity under iteration of an arbitrary exponential map. Providing exact formulas, we show how sensitively the Hausdorff dimension depends on the rate of growth of canonical Devaney–Krych codes.

We prove that simple, thick hyperbolic P-manifolds of dimension at least three exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension at least three. The key tool in the proof of these rigidity results is a strong form of the Jordan separation theorem, for maps from $S^n\rightarrow S^{n+1}$ which are not necessarily injective.

We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip $0<\Re(z)<1$. We study some of the properties of these functions, including their oscillatory behaviour.

Let $f\in \mathbb{Z}[ x,y]$ be a reducible homogeneous polynomial of degree 3. We show that $f(x,y)$ has an even number of prime factors as often as an odd number of prime factors.

In this paper, we prove that positive conditional entropy implies the existence of asymptotic pairs and scrambled sets on fibers. Moreover, we introduce the notion of conditional topological entropy for a subset using Bowen's definition of separated and spanning sets, and prove that the conditional topological entropy of a system relative to a factor is the supremum of conditional topological entropy of its scrambled sets on fibers.

We say that a finite abelian group does not have the Rédei property if it can be expressed as a direct product of two of its subsets such that both subsets contain the identity element and both subsets span the whole group. It will be shown that only a small fraction of the finite abelian groups can have the Rédei property. For groups of odd order an explicit list of the possible exceptions is compiled.

We investigate the limit functions of iterates of a function belonging to a convergence group or of a uniformly quasiregular mapping. We show that it is not possible for a subsequence of iterates to tend to a non-constant limit function, and for another subsequence of iterates to tend to a constant limit function. It follows that the closure of the stabiliser of a Siegel domain for a uniformly quasiregular mapping is a compact abelian Lie group, which we further conjecture to be infinite. This result concerning possible limits of convergent subsequences of iterates for holomorphic rational functions on the Riemann sphere is known, and the only known method of proof involves universal covering surfaces and Möbius groups. Hence, our method yields a new and perhaps more elementary proof also in that case.

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in three complex dimensions.

For a variety X of dimension n in ${\mathbb P}^r,\ r\geq n(k+1)+k$, the kth secant order of X is the number $\mu_k(X)$ of $(k+1)$-secant k-spaces passing through a general point of the kth secant variety. We show that, if $r>n(k+1)+k$, then $\mu_k(X)=1$ unless X is k-weakly defective. Furthermore we give a complete classification of surfaces $X\subset{\mathbb P}^r,\ r>3k+2$, for which $\mu_k(X)>1$.

We consider singular exact Poisson structures on an oriented manifold of dimension 4. Using normal forms for divergence-free vector fields on ${\mathbb R}^3$ and for 1-parameter deformations of functions on ${\mathbb R}^3$ we produce normal forms for two classes of such Poisson structures around the singular point.

Poincaré's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms homotopic to the identity, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempted classification, and in fact we define a system as $\rho$-bounded if such deviations are bounded and as $\rho$-unbounded otherwise. For the $\rho$-bounded case we prove a close analogue of Poincaré's result: if the rotation number is rationally related to the rotation rate on the base then there exists an invariant strip (the appropriate analogue for fixed or periodic points in this context), otherwise the system is semi-conjugate to an irrational translation of the torus. In the $\rho$-unbounded case, where neither of these two alternatives can occur, we show that the dynamics are always topologically transitive.

Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square correspond to the sextonions. This is a six-dimensional subalgebra of the split octonions which contains the split quaternions.

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.

We show that for every natural number m a finitely generated metabelian group G embeds in a quotient of a metabelian group of type $\textit{FP}_m$. Furthermore, if $m \leq 4$, the group G can be embedded in a metabelian group of type $\textit{FP}_m$. For L a finitely generated metabelian Lie algebra over a field K and a natural number m we show that, provided the characteristic p of K is 0 or $p > m$, then L can be embedded in a metabelian Lie algebra of type $\textit{FP}_m$. This result is the best possible as for $0 < p\leq m$ every metabelian Lie algebra over K of type $\textit{FP}_m$ is finite dimensional as a vector space.

Let M be a Hamiltonian K-space with proper moment map $\mu$. The symplectic quotient $X=\mu^{-1}(0)/K$ is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of $\mu^{-1}(0)$ to the middle perversity intersection cohomology of X, to this symplectic setting.

The key technical results which allow us to do this are Meinrenken's and Sjamaar's partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X.

Criteria for selfadjointness of integral operators, including matrix operators, based on our earlier domination results are established. The range of applicability is elucidated by carefully chosen examples, not covered by previously known criteria.