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 $g$.

In some recent papers, the authors considered regular continued fractions of the form \[ \bigg[a_{0};\underbrace{a,\ldots, a}_{m}, \underbrace{a^{2},\ldots, a^{2}}_{m}, \underbrace{a^{3},\ldots, a^{3}}_{m}, \ldots \bigg], \] where $a_{0} \,{\geq}\, 0, a \,{\geq}\, 2$ and $m \,{\geq}\, 1$ are integers. The limits of such continued fractions, for general $a$ and in the cases $m\,{=}\,1$ and $m\,{=}\,2$, were given as ratios of certain infinite series.

However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases $m\,{=}\,1$ and $m\,{=}\,2$ and also use known results about other $q$-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions.

Considérons la fonction arithmétique $$D^*(n,t)\colonequal \sum_{\di{d|n,\,d'|n}{0<|\log (d'/d)|\le t}} 1\qquad (n\,{>}\,1,\,t\,{>}\,0),$$ qui généralise la fonction $T^*(n,\alpha)\colonequal D^*(n,(\log n)^{-\alpha})$, étudiée dans la première partie de cet article [7]. Les variations de $D^*(n,t)$ au voisinage de l'origine permettent de mesurer, en moyenne quadratique, la propension des diviseurs de l'entier $n$ à s'agglutiner autour de certaines valeurs. Nous nous proposons ici de donner une majoration essentiellement optimale de cette fonction de $t$ sur $[0,1]$ lorsque $n$ est astreint à parcourir un ensemble d'entiers non exceptionnels en un certain sens quantitativement précisé.

Bescovitch considered the Hausdorff dimensions of sets related to the sum of digits of real numbers represented in the dyadic expansions. In this paper, we provide an analogy for continued fraction expansions over the field of formal Laurent series. We also calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations converge faster than those of almost all Laurent series (with respect to the Haar measure).

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.

Lejeune–Jalabert and Reguera computed the geometric Poincaré series $P_{geom}(T)$ for toric surface singularities. They raised the question whether this series equals the arithmetic Poincaré series. We prove this equality for a class of toric varieties including the surfaces, and construct a counterexample in the general case. We also compute the motivic Igusa Poincaré series $Q_{geom}(T)$ for toric surface singularities, using the change of variables formula for motivic integrals, thus answering a second question of Lejeune–Jalabert and Reguera's. The series $Q_{geom}(T)$ contains more information than the geometric series, since it determines the dimension of the tangent space at the singularity. In some sense, this is the only difference between $Q_{geom}(T)$ and $P_{geom}(T)$.

The following theorem is proved. For any positive integers $k$ and $n$ there exists $t$ depending only on $k$ and $n$ such that the class of all groups $G$ having $\gamma_k(G)$ locally finite and satisfying the condition that the product of any $t$ commutators of the form $[x_1,x_2,{\dots}\,,x_k]$ is of order dividing $n$ is a variety.

Two spatial embeddings of a graph are said to be delta edge-homotopic if they are transformed into each other by self delta moves and ambient isotopies. In this paper we classify theta curves up to delta edge-homotopy in terms of the third coefficient of the Conway polynomial of an associated 2-component link. In particular, we show that every boundary theta curve is delta edge-homotopically trivial, and two cobordant theta curves are delta edge-homotopic.

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.

J. Dean introduced the notion of primitive/Seifert-fibered knots and constructions to explain how Dehn surgeries on knots produce small Seifert fibered manifolds. We determine non-hyperbolic, primitive/Seifert-fibered knots, and show that for such knots any integral, small Seifert fibered surgery arises from a primitive/Seifert-fibered construction. We also show that any 2-orbifold $S^2(n_1, n_2, n_3)$ with a pair of indices coprime becomes the base orbifold of a Seifert fibered manifold obtained by Dehn surgery on a hyperbolic, primitive/Seifert-fibered knot in $S^3$.

In this paper, we shall prove that if $G$ is a countably infinite product of second countable amenable locally compact groups $G_i,i\,{=}\,0,1,2,\dots$ and each $G_i$ is a non-trivial compact group for $i>0,$ then the center of the second conjugate algebra of the Fourier algebra $A(G)$ of $G$ is precisely $A(G)$.

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.

Throughout we let ($T,{\frak m},k$) denote a commutative Noetherian local ring with maximal ideal ${\frak m}$ and residue field $k$. We let $I\,{\subseteq}\,T$ be an ideal generated by a regular sequence of length $c$ and set $R\,{\colonequal}\,T/I$. In the important paper [1], Avramov addresses the following question. Given a finitely generated $R$-module $M$, when does $M$ have finite projective dimension over a ring of the form $T/J$, where $J$ is generated by part (or all) of a set of minimal generators for $I$? He gives a fairly complete answer to this question that is expressed in terms of the geometry of varieties in affine space defined by annihilators of certain graded modules derived from resolutions over $R$. In an attempt to understand these ideas more fully, we became interested in the idea that one might answer the question at hand by using data about $M$ (orits syzygies) coming from $T$, in particular, information gleaned from various Fitting ideals defined over $T$. The following theorem from Section 3 is one of our main results. We use $\Fitt_T(M)$ to denote the Fitting ideal of $M$.

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 study eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha–Ko. We give a formula for the eta-invariant for boundary links. In several interesting cases this allows us to show that a given link is not slice. We show that even more eta-invariants have to vanish for boundary slice links.

We give an elegant simple proof of a result concerning the first normal bundle of ideal Lagrangian immersions. This enables us to simplify and clarify several of the results in an earlier paper.

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 $S_3*_{Z/2Z}S_3$ satisfies Turaev's criterion to be the fundamental group of an orientable $PD_3$-complex. Since this group has infinitely many ends but is indecomposable as a free product this provides a negative answer to a question of Wall.

We prove that the Casson–Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot $K$, all but finitely many algebraically slice twisted doubles of $K$ are linearly independent in the knot concordance group.

A point in Baire space is found for which the first derived $\omega$-limit set is not Borel, whilst the second is empty. A second point is found for which the sequence of derived $\omega$-limit sets does not stabilise until the first uncountable ordinal. The two points are recursive.