Explore the vast range of titles available.

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface${\\rm{S}}_{\\rm{g}}$of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of${\\rm{S}}_{\\rm{g}}$tends to zero at the rate 1/g. We consider here the smallest dilatation of any pseudo-Anosov homeomorphism of${\\rm{S}}_{\\rm{g}}$acting trivially on$\\Gamma /\\Gamma _k $, the quotient of$\\Gamma \\, = \\,\\pi _1 (S_g )$by the${\\rm{K}}^{{\\rm{th}}}$term of its lower central series, k > 1. In contrast to Penner's asymptotics, we prove that this minimal dilatation is bounded above and below, independently of g, with bounds tending to infinity with k. For example, in the case of the Torelli group${\\rm{I(S}}_g )$, we prove that${\\rm{L(I(S}}_g ))$, the logarithm of the minimal dilatation in${\\rm{I(S}}_g )$, satisfies .197 <${\\rm{L(I(S}}_g ))$< 4.127. In contrast, we find pseudo-Anosov mapping classes acting trivially on$\\Gamma /\\Gamma _k $whose asymptotic translation lengths on the complex of curves tend to 0 as g → ∞.

Consider the algebraic function $\\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

We prove a new structural result for the spherical Tits building attached to ${\\rm SL}_n K$ for many number fields $K$, and more generally for the fraction fields of many Dedekind domains ${\\cal O}$: the Steinberg module ${\\rm St}_n(K)$ is generated by integral apartments if and only if the ideal class group ${\\rm cl}({\\cal O})$ is trivial. We deduce this integrality by proving that the complex of partial bases of ${\\cal O}^n$ is Cohen-Macaulay. We apply this to prove new vanishing and non-vanishing results for ${\\rm H}^{\\nu_n}({\\rm SL}_n{\\cal O}_K;{\\Bbb Q})$, where ${\\cal O}_K$ is the ring of integers in a number field and $\\nu_n$ is the virtual cohomological dimension of ${\\rm SL}_n{\\cal O}_K$. The (non)vanishing depends on the (non)triviality of the class group of ${\\cal O}_K$. We also obtain a vanishing theorem for the cohomology ${\\rm H}^{\\nu_n}({\\rm SL}_n{\\cal O}_K;V)$ with twisted coefficients $V$.

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and holomorphic aspects, and so they also provide motivation for natural problems in geometric group theory and topology.

This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on \\mathbf R with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in \\mathrm{Diff}^2(\\mathbf R) as those groups whose elements have at most one fixed point.

Let \\({\\mathcal M}_{g,n}\\) denote the moduli space of smooth, genus \\(g\\geq 1\\) curves with \\(n\\geq 0\\) marked points. Let \\({\\mathcal A}_h\\) denote the moduli space of \\(h\\)-dimensional, principally polarized abelian varieties. Let \\(g\\geq 3\\) and \\(h\\leq g\\). If \\(F:{\\mathcal M}_{g,n}\\to{\\mathcal A}_h\\) is a nonconstant holomorphic map then \\(h=g\\) and \\(F\\) is the classical period mapping, assigning to a Riemann surface \\(X\\) its Jacobian.

This is a paper in smooth \\(4\\)-manifold topology, inspired by the Mordell-Weil Theorem in number theory. More precisely, we prove a smooth version of the Mordell-Weil Theorem and apply it to the `unipotent radical' case of a Thurston-type classification of mapping classes of simply-connected \\(4\\)-manifolds \\(M_d\\) that admit the structure of an elliptic complex surface of arithmetic genus \\(d\\geq 1\\). Applications include Nielsen realization theorems for \\(M_d\\). By combining this with known results, we obtain the following remarkable consequence: if the singular fibers of such an elliptic fibration are of the simplest (i.e.\\ nodal) type, then the fibered structure is unique up topological isotopy. In particular, any diffeomorphism of \\(M_d,d\\geq 3\\) is topologically isotopic to a diffeomorphism taking fibers to fibers.