We evaluate the character of the reflection representation of the split special orthogonal group over a finite field $F$ of odd characteristic. The value of the character at $g$ is expressed in terms of the dimensions of the eigenspaces of $g$ for eigenvalues in $F$ and in the norm 1 torus in the quadratic extension of $F$, and the behavior of the restriction of the quadratic form preserved by the group to the 1 and ${-}1$ eigenspaces of $g$.

We determine those pseudovarieties of groups ${\bf H}$ for which the power monoids $P(G)$, ranging over all groups $G$ in ${\bf H}$, satisfy the same profinite identities (i.e. pseudoidentities) as all semidirect products of ${\cal J}$-trivial monoids by groups in ${\bf H}$. That is, in the language of finite monoid theory, we characterize all solutions to the pseudovariety equation ${\bf PH}\,{=}\,{\bf J}\ast {\bf H}$. The characterization is in terms of the geometry of the Cayley graphs of the free pro-${\bf H}$ groups as well as in terms of the pro-${\bf H}$ topology of a finitely generated free group.

This is a brief study of the homology of cubical sets, with two main purposes.

First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes.

But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologically-trivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of ‘noncommutative topology’, without the metric information of C*-algebras.

We work on based CW complexes of finite types. In this paper, we show that each $p$-complete $H$-space can be uniquely decomposed into the product of atomic spaces and that each $p$-complete suspension space can be written as the $p$-completion of the wedge of atomic spaces. We then discuss some related problems on atomic spaces.

This paper uses some recent calculations of Conder and Bujalance (on classifying finite index group extensions of Fuchsian groups with abelian quotient and torsion free kernel) in order to determine the full automorphism groups of some cylic coverings of the line (including curves of Fermat and Lefschetz type). The answer is complete for cyclic covers that branch over three points.

We show that for every $g\geq 2$ there is a compact arithmetic Riemann surface of genus $g$ with at least $4(g-1)$ automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24.

We show that for every $n\,{\geq}\,2$ there exists closed hyperbolic n-manifolds for which the full group of orientation preserving isometries is trivial.

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. We also give a general formula for the Poincaré series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.

The main result of this paper gives an explicit computation of the $L_4$ norm of any cyclotomic polynomial of the form \[ f(x)=\Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1 p_2\cdots p_{r-1}}). \] Here $\Phi_{p}$ is the $p$th cyclotomic polynomial and the $p_i$ are primes that are not necessarily distinct.

A corollary of this is the following theorem.

THEOREM. If\[ f(x)=\Phi_{p_1}(\e_1 x)\Phi_{p_2}(\e_2 x^{p_1})\cdots \Phi_{p_r}(\e_r x^{p_1 p_2\cdots p_{r-1}}), \]where N = p1p2···prand εj = ±, then \[ \frac{\| f\|_4^4}{N^2} & \ge & \frac{\big\|\Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big) \big\|_4^4}{4^r}\\& = & \frac{\big(\frac12 +\frac{5}{34}\sqrt{17}\big)(1+\sqrt{17})^r - \big({-}\frac12 +\frac{5}{34}\sqrt{17}\big)(1-\sqrt{17})^r}{4^r}. \] In particular the minimum possible normalized L4 norm of any polynomial of the above form is attained by \[ \Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big). \]

Cochran, Orr and Teichner introduced $L^2$-eta-invariants to detect highly non-trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$-eta-invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta-invariants. We show that if for a knot $K$ this ribbon obstruction vanishes then the metabelian $L^2$-eta-invariant vanishes too. The converse has been shown by the author not to be true.

We study the modified Helmholtz equation in a semi-strip with Poincaré type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a $2\times 2$-matrix Riemann–Hilbert (RH) problem. If the six parameters specified by the boundary conditions satisfy certain algebraic relations this RH problem can be solved in closed form. For certain values of the parameters the solution is not unique, furthermore in some cases the solution exists only under certain restrictions on the functions specifying the boundary conditions. The asymptotics of the solution at the corners of the semi-strip is investigated. In the case that the $2\times 2$ RH problem cannot be solved in closed form, the Carleman–Vekua method for regularising it is illustrated by analysing in detail a particular case.