On differentiable compactifications of the hyperbolic space

The group of direct isometries of the hyperbolic space This isometric action admits many differentiable compactifications into an action on the closed n-dimensional ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the classifications of real analytic and smooth compactifications.     

Potential of the Weil-Petersson metric on the Torelli space

It is proved that the function12 π log (Z′(1)/det Im τ), where Z(s) is Sel'berg zeta function and τ is the matrix of the periods of a distinguished Riemann surface, is a potential of the Weil-Petersson metric on the Torelli space. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 110–120, 1987.     

Extending a convexity space to an aligned space

An aligned space — introduced by R.E. Jamison — is a convexity space with the additional property of domain finiteness. It is shown in this paper that each convexity space can be extended to a minimal domain finite convexity space, i.e. a minimal aligned space. Also a characterization of this extension is given.     

Alternate compactifications of the moduli space of genus one maps

We extend the definition of an m-stable curve introduced by Smyth to the setting of maps to a projective variety X, generalizing the definition of a Kontsevich stable map in genus one. We prove that the moduli problem of n-pointed m-stable genus one maps of class β is representable by a proper Deligne–Mumford stack ${\overline{\mathcal {M}}_{1,n}^{m}(X,\beta)}$ over Spec ${\mathbb {Z}[1/6]}$ . For ${X=\mathbb {P}^{r},}$ we show that ${\overline{\mathcal {M}}_{1,n}^{m}(\mathbb {P}^{r},d)}$ is irreducible for m sufficiently large. We also show that ${\overline{\mathcal {M}}_{1,n}^{m}(\mathbb {P}^r,d)}$ is smooth if d?+?n ≤ m ≤ 5.     

The p-rank stratification on the Siegel moduli space with Iwahori level structure

Our concern in this paper is to describe the p-rank stratification on the Siegel moduli space with Iwahori level structure over fields of positive characteristic. We calculate the dimension of the strata and describe the closure of a given stratum in terms of p-rank strata. We also examine the relationship between the p-rank stratification and the Kottwitz–Rapoport stratification.     

Canonical extensions of the Johnson homomorphisms to the Torelli groupoid

We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical extensions of the higher Johnson homomorphisms τm, for m?1, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these extensions. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which extends τ₂ and thus answer affirmatively a question of Morita and Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating τm, for m?3.     

A new compactification of the Siegel space and degeneration of Abelian varieties. II

This is a continuation of the article with the same title appeared in this journal, to the introduction of which we referr the reader for the summary of contents of this article.Dedicated to Professor K. Kodaira on his 60th birthday.     

Cohomology of the hyperelliptic Torelli group

Let SI(S _g ) denote the hyperelliptic Torelli group of a closed surface S _g of genus g. This is the subgroup of the mapping class group of S _g consisting of elements that act trivially on H ₁(S _g ; ?) and that commute with some fixed hyperelliptic involution of S _g . We prove that the cohomological dimension of SI(S _g ) is g ? 1 when g ≥ 1. We also show that H _g?1(SI(S _g ); ?) is infinitely generated when g ≥ 2. In particular, SI(S ₃) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B _n at t = ?1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.     

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

An Infinite Presentation of the Torelli Group

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair maps”, and all “commutators of simply intersecting pairs” and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of all simple closed curves.     

On properties of infimal topology of a map space

We study properties of the infimal topology τ_inf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τ_inf to have the Fréchet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τ_μ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τ_inf to satisfy the first axiom of countability.