We give a complete condition for any $n$ elements of ${\rm PSL}(2, {\mathbb R})$ to generate a Fuchsian group which is a Schottky group on that set of generators, and apply it to a question of Bowditch on representations of surface groups into ${\rm PSL}(2, {\mathbb R})$.

2000 Mathematical Subject Classification: 20H10, 32G15.

The set of normalizers between von Neumann (or, more generally, reflexive) algebras $\mathcal{A}$ and $\mathcal{B}$ (that is, the set of all operators $T$ such that $T \mathcal{A} T^{\ast} \subseteq \mathcal{B}$ and $T^{\ast} \mathcal{B} T \subseteq \mathcal{A}$) possesses ‘local linear structure’: it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces $\mathcal{U}$ of operators satisfying $\mathcal{UU}^{\ast} \mathcal{U} \subseteq \mathcal{U}$ (also known as ternary rings of operators). Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form $\mathcal{U} = \{T : TL = \phi (L) T$ for all $L \in \mathcal{L}\}$ where $\mathcal{L}$ is a set of projections and $\phi$ a certain map defined on $\mathcal{L}$. A normalizing space consists of normalizers between appropriate von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian self-adjoint algebras consist of operators ‘supported’ on sets of the form $[f = g]$ where $f$ and $g$ are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.

2000 Mathematical Subject Classification: 47L05 (primary), 47L35, 46L10 (secondary).

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well-known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest.

For any non-empty subset $\mathcal I$ of the natural numbers, let $\Lambda_{\mathcal I}$ denote those numbers in the unit interval whose continued fraction digits all lie in $\mathcal I$. Define the corresponding transfer operator

$$ \mathcal L_{\mathcal I, \beta} f(z) = \sum_{n \in \mathcal I} \left( \frac{1}{n + z} \right)^{2 \beta} f\left( \frac{1}{n + z} \right) $$

for $\text{Re} (\beta) > \max (0, \theta_{\mathcal I})$, where $\text{Re} (\beta) = \theta_{\mathcal I}$ is the abscissa of convergence of the series $\sum_{n \in \mathcal I}n^{-2 \beta}$.

When acting on a certain Hilbert space $\mathcal H_{\mathcal I, \beta}$, we show that the operator $\mathcal L_{\mathcal I, \beta}$ is conjugate to an integral operator $\mathcal K_{\mathcal I, \beta}$. If furthermore $\beta$ is real, then $\mathcal K_{\mathcal I, \beta}$ is selfadjoint, so that $\mathcal L_{\mathcal I, \beta} : \mathcal H_{\mathcal I, \beta} \to \mathcal H_{\mathcal I, \beta}$ has purely real spectrum. It is proved that $\mathcal L_{\mathcal I, \beta}$ also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space $C^\omega [0, 1]$, and on the Fréchet space $C^\infty [0, 1]$.

The analytic properties of the map $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ are investigated. For certain alphabets $\mathcal I$ of an arithmetic nature (for example, $\mathcal I = \{\text{primes}\}$, $\mathcal I = \{\text{squares}\}$, $\mathcal I$ an arithmetic progression, $\mathcal I$ the set of sums of two squares) it is shown that $\beta \mapsto \mathcal L_{\mathcal I, \beta}$ admits an analytic continuation beyond the half-plane $\text{Re} (\beta) > \theta_{\mathcal I}$.

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.

2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

In this paper the spaces of algebraic cycles on a real projective variety $X$ are studied as $\mathbb{Z}/2$-spaces under the action of the Galois group ${\rm Gal}(\mathbb{C}/\mathbb{R})$. In particular, the equivariant homotopy type of the group of algebraic $p$-cycles $\mathcal{Z}_p(\mathbb{P}_{\mathbb{C}}^n)$ is computed. A version of Lawson homology for real varieties is proposed. The real Lawson homology groups are computed for a class of real varieties.

2000 Mathematical Subject Classification: primary 55P91; secondary 14C05, 19L47, 55N91.

We derive absolute stability results of Popov-type for infinite-dimensional systems in an input-output setting. Our results apply to feedback systems where the linear part is the series interconnection of an $L^2$-stable linear system and an integrator, and the non-linearity satisfies a sector condition which allows for non-linearities with lower gain equal to zero (such as saturation, or more generally, bounded non-linearities). These results are used to prove convergence and stability properties of low-gain integral feedback control applied to $L^2$-stable linear systems subject to actuator and sensor non-linearities. The class of actuator/sensor non-linearities under consideration contains standard non-linearities which are important in control engineering such as saturation and deadzone. Moreover, we use the input-output theory developed to derive state-space results on absolute stability and low-gain integral control for strongly stable well-posed infinite-dimensional linear systems.