A regular map of type $\{m,n\}$ is a 2-cell embedding of a graph in an orientable surface, with the property that for any two directed edges $e$ and $e'$ there exists an orientation-preserving automorphism of the embedding that takes $e$ onto $e'$, and in which the face length and the vertex valence are $m$ and $n$, respectively. Such maps are known to be in a one-to-one correspondence with torsion-free normal subgroups of the triangle groups $T(2,m,n)$. We first show that some of the known existence results about regular maps follow from residual finiteness of triangle groups. With the help of representations of triangle groups in special linear groups over algebraic extensions of ${\mathbb Z}$ we then constructively describe homomorphisms from $T(2,m,n)=\langle y,z|\ y^m=z^n=(yz)^2=1\rangle$ into finite groups of order at most $c^r$ where $c=c(m,n)$, such that no non-identity word of length at most $r$ in $x,y$ is mapped onto the identity. As an application, for any hyperbolic pair $\{m,n\}$ and any $r$ we construct a finite regular map of type $\{m,n\}$ of size at most $C^r$, such that every non-contractible closed curve on the supporting surface of the map intersects the embedded graph in more than $r$ points. We also show that this result is the best possible up to determining $C=C(m,n)$. For $r\ge m$ the graphs of the above regular maps are arc-transitive, of valence $n$, and of girth $m$; moreover, if each prime divisor of $m$ is larger than $2n$ then these graphs are non-Cayley. 2000 Mathematics Subject Classification: 05C10, 05C25, 20F99, 20H25.

As is usual in prime number theory, write $$\psi(x,q,a)=\sum_{{n\le x}\atop{n\equiv a\pmod q}}\Lambda(n).$$ It is well known that when $q$ is close to $x$ the average value of $$V(x,q)=\sum_{{a=1}\atop{(a,q)=1}}^q \left|\psi(x,q,a)-{x\over{\phi(q)}}\right|^2$$ is about $x\log q$, and recently Friedlander and Goldston have shown that if $$U(x,q)=x\log q-x\left( \gamma+\log 2\pi+\sum_{p|q}\frac{\log p}{p-1} \right),$$ then the first moment of $V(x,q)-U(x,q)$ is small. In this memoir it is shown that the same is true for all moments. 2000 Mathematics Subject Classification: 11N13.

The largest prime factor of $X^{3}+2$ was investigated in 1978 by Hooley, who gave a conditional proof that it is infinitely often at least as large as $X^{1+\delta}$, with a certain positive constant $\delta$. It is trivial to obtain such a result with $\delta=0$. One may think of Hooley's result as an approximation to the conjecture that $X^{3}+2$ is infinitely often prime. The condition required by Hooley, his R$^{*}$ conjecture, gives a non-trivial bound for short Ramanujan--Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of $X^{3}+2$ is infinitely often at least as large as $X^{1+\delta}$, though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non-trivial bound for short Ramanujan--Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus. 2000 Mathematics Subject Classification: 11N32.

Given any sequence of non-abelian finite simple primitive permutation groups $(S_{n})$, we construct a finitely generated group $G$ whose profinite completion is the infinite permutational wreath product $\ldots S_{n}\wr S_{n-1}\wr\ldots\wr S_{0}$. It follows that the upper composition factors of $G$ are exactly the groups $S_{n}$. By suitably choosing the sequence $(S_{n})$ we can arrange that $G$ has any one of a continuous range of slow, non-polynomial subgroup growth types. We also construct a $61$-generator perfect group that has every non-abelian finite simple group as a quotient. 2000 Mathematics Subject Classification: 20E07, 20E08, 20E18, 20E32.

Let $G$ be an almost simple algebraic group defined over ${\Bbb F}_p$ for some prime $p$. Denote by $G_1$ the first Frobenius kernel in $G$ and let $T$ be a maximal torus. In this paper we study certain Jantzen type filtrations on various modules in the representation theory of $G_1T$. We have such filtrations on the baby Verma modules $Z(\lambda)$, where $\lambda$ is a character of $T$. They are obtained via a certain deformation of the natural homomorphism from $Z(\lambda)$ into its contravariant dual $Z(\lambda)^\tau$. Using the same deformation we construct for each projective $G_1T$-module $Q$ a filtration of the vector space $F_\lambda(Q)=\text{Hom}_{G_1T}(Z(\lambda)^\tau, Q)$. We then prove that this filtration may also be described in terms of the above-mentioned homomorphism $Z(\lambda) \rightarrow Z(\lambda)^\tau$ and this leads us to a sum formula for our filtrations. When $Q$ is indecomposable with highest weight in the bottom alcove (with respect to some special point) we are able to compute the filtrations on $F_\lambda(Q)$ explicitly for all $\lambda$. This is then the starting point of an induction which proceeds via wall crossings to higher alcoves. If our filtrations behave as expected under such wall crossings then we obtain a precise relation between the dimensions of the layers in the filtrations of $F_\lambda (Q)$ for an arbitrary indecomposable projective $Q$ and the coefficients in the corresponding Kazhdan--Lusztig polynomials. We conclude the paper by proving that the above results in the $G_1T$ theory have some analogues in the representation theory of $G$ (where, however, we have to work with representations of bounded highest weights) and the corresponding theory for quantum groups at roots of unity. These results extend previous work by the first author. 2000 Mathematics Subject Classification: 20G05, 20G10, 17B37.

Let $G$ be a semisimple algebraic group defined over an algebraically closed field $K$ of good characteristic $p>0.$ Let $u$ be a unipotent element of $G$ of order $p^{t}$, for some $t\in {\Bbb N}$. In this paper it is shown that $u$ lies in a closed subgroup of $G$ isomorphic to the {\it Witt group} $W_{t}(K),$ which is a $t$-dimensional connected abelian unipotent algebraic group. 2000 Mathematics Subject Classification: 20G15.

Let $G$ be a $\sigma$-compact, locally compact group and $\mathcal I$ be a closed 2-sided ideal with finite codimension in $L^1(G)$. It is shown that there are a closed left ideal ${\mathcal L}$ having a right bounded approximate identity and a closed right ideal ${\mathcal R}$ having a left bounded approximate identity such that ${\mathcal I} = {\mathcal L} + {\mathcal R}$. The proof uses ideas from the theory of boundaries of random walks on groups. 2000 Mathematics Subject Classification: primary 43A20; secondary 42A85, 43A07, 46H10, 46H40, 60B11.

We explicitly determine the high-energy asymptotics for Weyl--Titchmarsh matrices corresponding to matrix-valued Schr\"odinger operators associated with general self-adjoint $m\times m$ matrix potentials $Q\in {L^1_{\text{loc}} ((x_0,\infty))}^{m\times m}$, where $m\in{\mathbb N}$. More precisely, assume that for some $N\in {\mathbb N}$ and $x_0\in{\mathbb R}$, $Q^{(N-1)}\in L^1([x_0,c))^{m\times m}$ for all $c>x_0$, and that $x\geq x_0$ is a right Lebesgue point of $Q^{(N-1)}$. In addition, denote by $I_m$ the $m\times m$ identity matrix and by $C_\varepsilon$ the open sector in the complex plane with vertex at zero, symmetry axis along the positive imaginary axis, and opening angle $\varepsilon$, with $0<\varepsilon< \frac12\pi$. Then we prove the following asymptotic expansion for any point $M_+(z,x)$ of the unique limit point or a point of the limit disk associated with the differential expression $-I_m\frac{d^2}{dx^2}+Q(x)$ in ${L^2((x_0,\infty))}^m$ and a Dirichlet boundary condition at $x=x_0$: \begin{equation} M_+(z,x)\underset{|z|\to\infty,\, z\in C_\varepsilon}{=} i I_m z^{1/2}+ \sum_{k=1}^N m_{+,k}(x)z^{-k/2}+ o(|z|^{-N/2}), \quad \text{where }N\in{\mathbb N}. \nonumber \end{equation} The expansion is uniform with respect to $\arg\,(z)$ for $|z|\to \infty$ in $C_\varepsilon$ and uniform in $x$ as long as $x$ varies in compact subsets of ${\mathbb R}$ intersected with the right Lebesgue set of $Q^{(N-1)}$. Moreover, the $m\times m$ expansion coefficients $m_{+,k}(x)$ can be computed recursively. Analogous results hold for matrix-valued Schr\"odinger operators on the real line. 2000 Mathematics Subject Classification: 34E05, 34B20, 34L40, 34A55.

In this paper we study the eigenvalues and eigenfunctions of metric measure manifolds. We prove that any eigenfunction is $C^{1,\alpha}$ at its critical points and $C^{\infty}$ elsewhere. Moreover, the eigenfunction corresponding to the first eigenvalue in the Dirichlet problem does not change sign. We also discuss the first eigenvalue, the Sobolev constants and their relationship with the isoperimetric constants. 2000 Mathematics Subject Classification: 47J05, 47J10, 53C60, 58E05, 58C40.

Given a fibred, compact, orientable 3-manifold with single boundary component, we show that a fibration with fiber surface of negative Euler characteristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration. These taut foliations extend to taut foliations in the corresponding surgery manifolds. 2000 Mathematics Subject Classification: primary 57M25; secondary 57R30.