We define a group theoretic criterion on a pro-$p$ group that guarantees the existence of an analytic structure over ${\mathbb Z}_p[[t]]$. This is a one-way analogue of Lazard's succesful answer to the $p$-adic version of Hilbert's Fifth Problem. In the course of the proof we define a $T$-map, a $T$-uniform group and set up a category equivalence between $T$-uniform pro-$p$ groups and powerful ${\mathbb Z}_p[[t]]$-Lie algebras.

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative $H$-spaces in the sense that any map $f\colon X\to Y$ to a homotopy associative, homotopy commutative $H$-space extends to a uniquely determined $H$-map $\overline{f}\colon U(X)\to Y$. Developing a method for recognising certain universal spaces, we show the existence of the universal space $F_2(n)$ of a certain three-cell complex $L$. Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the $d_1$-differential of the EHP-spectral sequence valid in a certain range.

Functional analytic properties of the Banach Algebras $A_p^r(G)$ such as Arens regularity, amenability, weak sequential completeness, sets of synthesis and strict containment for fixed $p$ and increasing $r$, are investigated.

It is found that $A_p^r(G)$, $1\leq r<\infty$, behave very differently to the Figa–Talamanca–Herz algebras $A_p(G)$.

In 1977 Grünbaum and Shephard described all possible 93 types of isohedral marked tilings of the plane; 46 of them are called basic, since their induced tile group is trivial. The aim of this paper is to give an algebraic description of all basic tilings. A purely algebraic characterization of the adjacency symmetries of tiles of the 46 basic tilings is presented. Moreover, 46 related abstract definitions of two-dimensional crystallographic groups supplement and extend those of the well-known book Generators and Relations for Discrete Groups by Coxeter and Moser.

It is well known that every orientation preserving homeomorphism of the plane $\mathbb{R}^2$ with a $k$-periodic orbit ${\cal O}$ ($k \geq 2$) necessarily has a fixed point (see [4] or [5], [7], [11]). It is then interesting to know if ${\cal O}$ is linked with one of these fixed points. The difficulty in answering this question depends a lot on the precise sense we give to the word “linked”. Before discussing briefly this point, let us observe this general problem has a very natural counterpart in the framework of orientation reversing homeomorphisms of the sphere $\mathbb{S}^2$. Indeed, every such homeomorphism possessing a $k$-periodic orbit ${\cal O}$ ($k\geq 3$) also has a 2-periodic orbit ([2]) and one can ask if ${\cal O}$ is linked with one of these 2-periodic orbits. The aim of this paper is to try to answer this question.

We investigate the behavior of the higher-order degrees, $\bar{\delta}_n$, of a finitely presented group $G$. These $\bar{\delta}_n$ are functions from $H^1(G;{\mathbb Z})$ to ${\mathbb Z}$ whose values are the degrees certain higher-order Alexander polynomials. We show that if ${\rm def}(G)\geq 1$ or $G$ is the fundamental group of a compact, orientable 3-manifold then $\bar{\delta}_n$ is a monotonically increasing function of $n$ for $n\geq1$. This is false for general groups. As a consequence, we show that if a 4-manifold of the form $X \times S^1$ admits a symplectic structure then X “looks algebraically like” a 3-manifold that fibers over $S^1$, supporting a positive answer to a question of Taubes. This generalizes a theorem of S. Vidussi and is an improvement on previous results of the author. We also find new conditions on a 3-manifold $X$ that will guarantee that the Thurston norm of $f^{\ast}(\psi)$, for $\psi \in H^1(X;{\mathbb Z})$ and $f{:}\,Y \rightarrow X$ a surjective map on $\pi_1$, will be at least as large the Thurston norm of $\psi$. When $X$ and $Y$ are knot complements, this gives a partial answer to a question of J. Simon.

More generally, we define $\Gamma$-degrees, $\bar{\delta}_\Gamma$, corresponding to a surjective map $G\twoheadrightarrow \Gamma$ for which $\Gamma$ is poly-torsion-free-abelian. Under certain conditions, we show they satisfy a monotonicity condition if one varies the group $\Gamma$. As a result, we show that these generalized degrees give obstructions to the deficiency of a group being positive and obstructions to a finitely presented group being the fundamental group of a compact, orientable 3-manifold.

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this paper provide new examples of non transversally simple knot types.

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier–Stieltjes transform of a discrete measure on the dual group.

We show that if $\varepsilon\,{<}\,\sqrt2$ then an $\varepsilon$-Kronecker set is $I_0$, but this is not true for at least one $\sqrt 2$-Kronecker set. $\varepsilon$-Kronecker sets in ${\mathbb Z}$ need not be finite unions of Hadamard sets. As with Sidon sets, $\varepsilon$-Kronecker sets with $\varepsilon\,{<}\,2$ do not contain arbitrarily long arithmetic progressions or large squares. When $\varepsilon\,{<}\,\sqrt 2$ they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier Stieltjes transform of a discrete measure on the dual group.

We show that products (sums) of $\varepsilon$-Kronecker sets can be all of the group if the number of terms is sufficiently large, but are shown to be $U_0$ sets (sets of uniqueness in the weak sense) if the number is small. Results about cluster points of products are extended from Hadamard to $\varepsilon$-Kronecker sets. One consequence of that is that finite unions of translates of a fixed $\varepsilon$-Kronecker set are $I_0$.

This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces $W(L_p,\ell_q)({\mathbb R}^d)$, Herz spaces and weighted versions of Feichtinger's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the $\theta$-means $\sigma_T^\theta f$ can be estimated pointwise by the Hardy–Littlewood maximal function. From this it follows that $\sigma_T^\theta f \,{\to}\, f$ a.e. for all $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$, hence $f\in L_p({\mathbb R}^d)$ for any $1\leq p\leq \infty$. Moreover, $\sigma_T^\theta f(x)$ converges to $f(x)$ at each Lebesgue point of $f\in L_1({\mathbb R}^d)$ (resp. $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$) if and only if the Fourier transform of $\theta$ is in a suitable Herz space. In case $\theta$ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for $\theta$ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.

We prove monadic second order limit laws for classes of logarithmic adequate relational structures by using probabilistic methods.

Using coupling techniques, we prove uniqueness in $G$-measures under a weak regularity condition and give estimates of the associated rates of convergence. We also show how to generate a random variable distributed according to the unique $G$-measure on cylinder sets for any fixed level of precision.