We complete the determination of how far convex maps can deform discs in each of the three classical geometries. The euclidean case was settled by Nehari in 1976, and the spherical case by Mejía and Pommerenke in 2000. We find the sharp bound on the Schwarzian derivative of a hyperbolically convex function and thus complete the hyperbolic case. This problem was first posed by Ma and Minda in a series of papers published in the 1980s. Mejía and Pommerenke then produced partial results and a conjecture as to the extremal function in 2000. Their function maps onto a domain bounded by two proper geodesic sides, a ‘hyperbolic strip’. Applying a generalization of the Julia variation and a critical Step Down Lemma, we show that there is an extremal function mapping onto a domain with at most two geodesic sides. We then verify using special function theory that, among the remaining candidates, the two-sided domain of Mejía and Pommerenke is in fact extremal. This correlates nicely with the euclidean and spherically convex cases in which the extremal is known to be a map onto a two-sided ‘strip’.

A notion of $L^2$-determinant is established generalizing $L^2$-torsion. It is shown that classical regularized determinants of elliptic operators converge to $L^2$-determinants in a tower of coverings. A product formula is proved, which provides an Euler-product expansion to regularized determinants, having generalized $L^2$-determinants as Euler-factors.

1991 Mathematics Subject Classification: 58G26, 58F20.

We study the rigidity problem for periodic orbits of (continuous) graph maps belonging to the same homotopy equivalence class. Since the underlying spaces are not necessarily homeomorphic, we define a new notion of pattern which enables us to compare periodic orbits of self-maps of homotopy-equivalent spaces. This definition unifies the known notions of pattern for other spaces. The two main results of the paper are as follows: given a free group endomorphism, we study the persistence under homotopy of the periodic orbits of its topological representatives, and in the irreducible case, we prove the minimality (within the homotopy class) of the set of periodic orbits of its efficient representatives.

Let $f:({\mathbb R}^n,0)\rightarrow(\Bbb R^p,0)$ be a $C^\infty$ map-germ. We define $f$ to be finitely, or $\infty$-, $\mathcal A$-determined, if there exists an integer $m$ such that all germs $g$ with $j^mg(0)=j^mf(0)$, or if all germs $g$ with the same infinite Taylor series as $f$, respectively, are $\mathcal A$-equivalent to $f$. For any integer $k$, $0\le k<\infty$, we can consider $\mathcal A$\rq s $C^k$ counterpart (consisting of $C^k$ diffeomorphisms) $\mathcal A^{(k)}$, and we can define the notion of finite, or $\infty$-,$\mathcal A^{(k)}$-determinacy in a similar manner. Consider the following conditions for a $C^{\infty}$ germ $f$: (a$_k$) $f$ is $\infty$-$\mathcal A^{(k)}$-determined, (b$_k$) $f$ is finitely $\mathcal A^{(k)}$-determined, (t) $m_n^{\infty}\theta(f)\subset tf(m_n\theta(n))+\omega f(m_p\theta(p))$, (g) there exists a representative $ f:U\rightarrow\Bbb R^p$ defined on some neighbourhood $U$ of 0 in $\Bbb R^n$ such that the multigerm of $f$ is stable at every finite set $S\subset U-\lbrace0\rbrace$, and (g$^\prime$) every $f^\prime $ with $j^\infty f^\prime(0)=j^\infty f(0)$ satisfies condition (g). We also define a technical condition which will imply condition $(\mbox{g})$ above. This condition is a collection of $p+1$ \Lojasiewicz inequalities which express that the multigerm of $f$ is stable at any finite set of points outside 0 and only becomes unstable at a finite rate when we approach 0. We will denote this condition by (e). With this notation we prove the following. For any $ C^\infty$ map germ $f:(\Bbb R^n,0)\rightarrow(\Bbb R^p,0)$ the conditions (e), (t), (g$'$) and ($\mbox{a}_{\infty}$) are equivalent conditions. Moreover, each of these conditions is equivalent to any of ($\mbox{a}_k$) $(p+1\le k<\infty)$, ($\mbox{b}_k$) $(p+1\le k<\infty)$.

1991 Mathematics Subject Classification: 58C27.

We give a complete condition for any $n$ elements of ${\rm PSL}(2, {\mathbb R})$ to generate a Fuchsian group which is a Schottky group on that set of generators, and apply it to a question of Bowditch on representations of surface groups into ${\rm PSL}(2, {\mathbb R})$.

2000 Mathematical Subject Classification: 20H10, 32G15.

The techniques and concepts we present are flags of regular schemes and their persistence under blow-up, the Gauss–Bruhat decomposition of the group of formal automorphisms of affine space, and coordinate-free initial ideals. All three are used to construct and study invariants for resolution of singularities.

In this paper the authors provide strong evidence for a conjecture of Chinburg in the context of abelian extensions $N/K$ of quadratic imaginary fields $K$ with group $G$. For abelian groups $G$, the conjecture specialises to the statement that Chinburg's third invariant $\Omega(N/K,3)=0 \in \mbox{Cl}({\Bbb Z}[G])$. We prove (Theorem 1) that if $K$ has class number 1 (and some technical conditions on the conductor of $N/K$ and its roots of unity hold) then $\Omega(N/K,3)$ lies in the kernel group $D({\Bbb Z}[G])$. For a special class of genus extensions of prime power degree we prove (Theorem 2) the full conjecture, that $\Omega(N/K,3)=0$. The proof technique for Theorems 1 and 2 uses Kolyvagin's Euler systems of units applied to certain explicit groups of elliptic units, which essentially determines the difference in Galois structure between units and ideal class groups over the maximal order ${\mathcal M}_G$ in${\Bbb Q}[G]$. This is made precise by using Hecke factorisation and working in the factorisability defect class group of the category of finitely generated ${\Bbb Z}[G]$-modules, which is isomorphic to $\mbox{Cl}({\mathcal M}_G)$. For Theorem 2, the fact that the class number of $N$ is coprime to $[N:K]$ in such genus extensions $N/K$ allows one to lift these results to $\mbox{Cl}({\Bbb Z}[G])$, after using cohomological calculations to check that (essentially) an explicit representative for a Tate sequence defining $\Omega (N/K,3)$ has the correct extension class (which is related to the fundamental class of $N/K$).

1991 Mathematics Subject Classification: 11R33.

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.

Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

Let $M_{g,k}$ be the moduli space of flat SU(2)-connections on a compact Riemann surface $\Sigma_g$ of genus $g \ge 2$ punctured in a point $p$ in a small open disc $D$, with holonomy $(-1)^k$ about $\partial D$. This paper gives an explicit decomposition of the rational intersection homology of $M_{g,k}$ into irreducible representations of the oriented mapping class group $\mbox{Map}^+(\Sigma_g)$ of $\Sigma_g$.

1991 Mathematics Subject Classification: primary 14D20; secondary 14H60, 55N33.

Arakelov and Faltings developed an admissible theory on {\it regular} arithmetic surfaces by using Arakelov canonical volume forms on the associated Riemann surfaces. Such volume forms are induced from the associated Kähler forms of the flat metric on the corresponding Jacobians. So this admissible theory is in the nature of Euclidean geometry, and hence is not quite compatible with the moduli theory of Riemann surfaces. In this paper, we develop a general admissible theory for arithmetic surfaces (associated with stable curves) with respect to any volume form. In particular, we have a theory of arithmetic surfaces in the nature of hyperbolic geometry by using hyperbolic volume forms on the associated Riemann surfaces. Our theory is proved to be useful as well: we have a very natural Weil function on the moduli space of Riemann surfaces, and show that in order to solve the arithmetic Bogomolov-Miyaoka-Yau inequality, it is sufficient to give an estimation for Petersson norms of some modular forms.

1991 Mathematics Subject Classification: 11G30, 11G99, 14H15, 53C07, 58A99.