Let ${\cal M} =\langle M,<,\ldots\rangle$ be alinearly ordered structure. We define ${\cal M}$ to be{\em o-minimal} if every definable subset of $M$ is a finite union of intervals. Classical examples are ordered divisible abelian groups and real closed fields. We prove a trichotomy theorem for the structure that an arbitraryo-minimal ${\cal M}$ can induce on a neighbourhood of any $a$in $M$. Roughly said, one of the following holds: \begin{enumerate}\item[(i)] $a$ is trivial (technical term), {\em or}\item[(ii)] $a$ has a convex neighbourhood on which ${\cal M}$ induces the structure of an ordered vector space, {\em or}\item[(iii)] $a$ is contained in an open interval on which ${\cal M}$ induces the structure of an expansion of a real closed field. \end{enumerate} The proof uses ‘geometric calculus’ which allows one to recover a differentiable structure by purely geometric methods.

1991 Mathematics Subject Classification: primary 03C45; secondary 03C52, 12J15, 14P10.

Let $G$ be a geometrically finite Kleinian group with parabolic elements and let $p$ be any parabolic fixed point of $G$. For each positive real $\tau$, let ${\cal W}_{p}(\tau)$ denote the set of limit points of $G$ for which the inequality $$ | x-g(p) | \leq |g^\prime(0)|^{\tau}$$ is satisfied for infinitely many elements $g$ in $G$. This subset of the limit set is precisely the analogue of the set of $\tau$-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the `size' of the set ${\cal W}_{p}(\tau)$ expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for $\tau \geq 1$,$$ \dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$$ where $\mbox{rk}(p)$ denotes the rank of the parabolic fixed point $p$.

1991 Mathematics Subject Classification: 11K55, 11K60, 11F99, 58D20.

We prove the quantum version - for Hecke algebras $H(A_n)$ of type $A$ at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This result describes the socle of the restriction of an irreducible $H(A_n)$-module to the subalgebra $H(A_{n-1})$. As a consequence, we describe the quantum version of the Mullineux involution describing the irreducible module obtained on twisting an irreducible module with the sign representation.

1991 Mathematics Subject Classification: 20C05, 20G05.

Here, $F$ denotes a non-Archimedean local field (with finite residue field) and $G$ the group of $F$-points of a connected reductive algebraic group defined over $F$. Let $\frak R(G)$ denote the category of smooth, complex representations of $G$. Let ${\cal B}(G)$ be the set of pairs $(L,\sigma)$, where $L$ is an $F$-Levi subgroup of $G$ and $\sigma$ is an irreducible supercuspidal representation of $L$, taken modulo the equivalence relation generated by twisting with unramified quasi characters and $G$-conjugacy. To $\frak s\in {\cal B}(G)$, one can attach a full (abelian) sub-category $\frak R^\frak s(G)$ of $\frak R(G)$; the theory of the Bernstein centre shows that $\frak R(G)$ is the direct product of these $\frak R^\frak s(G)$. The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix $\frak s\in {\cal B}(G)$. Let $K$ be a compact open subgroup of $G$ and $\rho$ an irreducible smooth representation of $K$. The pair $(K,\rho)$ is an $\frak s$-type if it has the following property: an irreducible representation $\pi$ of $G$ contains $\rho$ if and only if $\pi\in \frak R^\frak s(G)$. Let ${\cal H}(G,\rho)$ be the Hecke algebra of compactly supported $\rho$-spherical functions on $G$; if $(K,\rho)$ is an $\frak s$-type, then the category $\frak R^\frak s(G)$ is canonically equivalent to the category ${\cal H}(G,\rho) \text{-Mod}$ of ${\cal H}(G,\rho)$-modules. Let $M$ be a Levi subgroup of $G$; there is a canonical map ${\cal B}(M)\to {\cal B}(G)$. Take $\frak t\in {\cal B}(M)$ with image $\frak s\in {\cal B}(G)$. The choice of a parabolic subgroup of $G$ with Levi component $M$ gives functors of {\it parabolic induction\/} and {\it Jacquet restriction\/} connecting $\frak R^\frak t(M)$ with $\frak R^\frak s(G)$. We assume given a $\frak t$-type $(K_M,\rho_M)$ in $M$; the paper concerns a general method of constructing from this data an $\frak s$-type $(K,\rho)$ in $G$. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism ${\cal H}(M,\rho_M) \to {\cal H}(G,\rho)$. The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing ${\cal H}(G,\rho)$ explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types.

1991 Mathematics Subject Classification: 22E50.

We study densities $\rho$ on the unit ball in euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure associated with $\rho$. For these densities a geometric theory can be developed which captures many features of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring-Hayman theorem, the radial limit theorem and find analogues of compression and expansion phenomena on the boundary.

1991 Mathematics Subject Classification: 30C65.

The invariantly harmonic functions in the unit ball ${\Bbb B}^n$ in ${\Bbb C}^n$ are those annihilated by the Bergman Laplacian $\Delta$. The Poisson-Szeg\"o kernel $P(z,\zeta)$ solves the Dirichlet problem for $\Delta$: if $f\in C(S^n)$, the Poisson-Szeg\"o transform of $f$, $P[f](z)=\int_{S^n}P(z,\zeta) f(\zeta)\,d\sigma(\zeta),$ where $d\sigma$ is the normalized Lebesgue measure on $S^n$, is the unique invariantly harmonic function $u$ in ${\Bbb B}^n$, continuous up to the boundary, such that $u=f$ on $S^n$. The Poisson-Szeg\"o transform establishes, loosely speaking, a one-to-one correspondence between function theory in $S^n$ and invariantly harmonic function theory in ${\Bbb B}^n$. When $n\geq 2$, it is natural to consider on $S^n$ function spaces related to its natural non-isotropic metric, for these are the spaces arising from complex analysis. In the paper, different characterizations of such spaces of smooth functions are given in terms of their invariantly harmonic extensions, using maximal functions and area integrals, as in the corresponding Euclidean theory. Particular attention is given to characterization in terms of purely radial or purely tangential derivatives. The smoothness is measured in two different scales: that of Sobolev spaces and that of Lipschitz spaces, including BMO and Besov spaces.

1991 Mathematics Subject Classification: 32A35, 32A37, 32M15, 42B25.

We study the global behaviour of trees of Markoff triples over the complex numbers. We relate this to the space of type-preserving representations of the punctured torus group into $\mbox{SL}(2,{\Bbb C})$. In particular, we explore which Markoff triples correspond to quasifuchsian representations. We derive a variation of McShane's identity for quasifuchsian groups. In the case of non-discrete representations, we attempt to relate the asymptotic behaviour of Markoff triples to the realisability of laminations in hyperbolic 3-space. We also consider how some of these issues might be related for more general surfaces.

1991 Mathematics Subject Classification: 57M50.