It is well known that some lattices in ${\rm SO}(n,1)$ can be nontrivially deformed when included in ${\rm SO}(n+1,1)$ (e.g., via bending on a totally geodesic hypersurface); this contrasts with the (super) rigidity of higher rank lattices. M. Kapovich recently gave the first examples of lattices in ${\rm SO}(3,1)$ which are locally rigid in ${\rm SO}(4,1)$ by considering closed hyperbolic $3$-manifolds obtained by Dehn filling on hyperbolic two-bridge knots. We generalize this result to Dehn filling on a more general class of one-cusped finite volume hyperbolic $3$-manifolds, allowing us to produce the first examples of closed hyperbolic $3$-manifolds which contain embedded quasi-Fuchsian surfaces but are locally rigid in ${\rm SO}(4,1)$.

A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. We characterize metric gauges that are locally branched Euclidean and discuss an obstruction to removing the branching. Our characterization is a mixture of analysis, geometry, and topology with an argument of Yu. Reshetnyak to produce the branched coordinates for the gauge.

We consider a function $\phi\in L\sp 2(\mathbb {R}\sp d)$ such that $\{| \det(D)|\sp {1/2}\phi(Dx-\lambda) : D\in \mathscr {D},\lambda\in \mathscr {T}\}$ forms an orthogonal basis for $L\sp 2(\mathbb {R}\sp d)$, where $\mathscr {D}\subset M\sb d(\mathbb {R})$ and $\mathscr {T}\subset \mathbb {R}\sp d)$. Such a function $\phi$ is called a wavelet with respect to the dilation set $\mathscr {D}$ and translation set $\mathscr {T}$. We study the following question: Under what conditions can a $\mathscr {D}\subset M\sb d(\mathbb {R}$ and a $\mathscr {T}\subset \mathbb {R}\sp d)$ be used as, respectively, the dilation set and the translation set of a wavelet? When restricted to wavelets of the form $\phi=\check{\chi}\Omega$, this question has a surprising tie to spectral sets and their spectra.

We use scattering theoretic methods to prove strong dynamical and exponential localization for one-dimensional, continuum, Anderson-type models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single-site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs.

In this paper we prove a new variant of the Herz majorizing principle for operators defined by $\mathbb {K}$-bi-invariant kernels with certain large-scale cancellation properties. As an application, we prove $L\sp p$-boundedness of operators defined by Fourier multipliers which satisfy singular differential inequalities of the Hörmander-Michlin type. We also find sharp bounds on the $L\sp p$-norm of large imaginary powers of the critical $L\sp p$-Laplacian.

In this paper we calculate the asymptotics of various moments of the central values of Rankin-Selberg convolution L-functions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexity-breaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators.