Curve Complexes Versus Tits Buildings: Structures and Applications

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat–Tits buildings. We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT ( 0 ) space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension ≥ 2 is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

Let G be a semisimple real Lie group of non-compact type, K a maximal compact subgroup and S G a semigroup with nonempty interior.We consider the ideal boundary (G/K) of the associated symmetric space and the flag manifolds G/P .We prove that the asymptotic imagewhere C is the maximal invariant control set for the action of S in the flag manifold G/P , with P a parabolic subgroup.The points that project over C are exactly the points of type in (Sx 0 ) (in the sense of the type of a cell in a Tits Building).

Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

An explicit bound is obtained for the torsion of elliptic curves over the field of rational numbers. Let Γ be an elliptic curve over the field of rational numbers R, and Qm a primitive R-point of order m on it; here m is a prime or a double prime. Hence if m=2p, then p ≤ 509, whereas if m=p, then p < 6144.

The classical problem of the periodicity of continued fractions for elements of hyperelliptic fields has a long and deep history. This problem has up to now been far from completely solved. A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomials with coefficients in the field of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over whose square root has a periodic continued fraction expansion in the field of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials with periodic continued fraction expansion of in elliptic fields with the field of rational numbers as the field of constants.

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof goes back to Minkowski and proceeds by reduction to the case of finite fields. For the most part, we work over an arbitrary number field rather than the rationals. The first half of the article is expository and is intended to be accessible to graduate students and advanced undergraduates. It gives a self-contained treatment, following Schur, over the field of rational numbers.

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I 2 is free A[T]/I-module of rank n and (A[T]/I) is torsion in K 0(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.

It is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children’s difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, in-depth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.

We determine the stabilityof totally geodesic submanifolds in a compact symmetric space, which are called polars and meridians (see 2.1). These subspaces were introduced by Chen and Nagano ([CN-1]) and we have many interesting results after that ([CN-2], [N-l], [N-2], [NS-1], [NS-2] and [NS-3]). Recently, several results have been obtained about the stabilityof totally geodesic submanifolds in compact symmetric spaces. Ohnita gave the formula for the index, the nullityand the Killing nullity of a totally geodesic submanifold in a compact symmetric space in [0], in which he also proved that the Helgason sphere in a compact symmetric space is stable. Tasaki proved that the Helgason sphere in a compact Lie group is homologically volume-minimizing in its real homology classin [Ts-1]. He used the calibrationtheory. And there are studies about the stability of certain closed subgroups in a compact Lie group by Mashimo and Tasaki ([MT-1] and [MT-2]). Mashimo determined all the unstable Cartan embeddings of compact symmetric spaces in [M]. And there is a result about the stabilityof symmetric i?-spaces in Hermitian symmetric spaces and totally complex submanifolds in quaternionic Kahler symmetric spaces of classical type by Takeuchi ([Tk-2]). Recently Nagano and the author have obtained a result on a relationshipbetween the stabilityof minimal submanifolds and that of /^-harmonic maps ([NS-3]). In the present paper we study the stability of all the polars and meridians in every compact symmetric space by using Ohnita's method in Section 3. We will also study the stabilityof totally complex totallygeodesic submanifolds in quaternionic Kahler symmetric spaces of exceptional type in Section 4. I should like to express my gratitude to Professor T. Nagano for his useful advice and kindly support. This is the author's doctoral dissertation submitted to Sophia University in October, 1992.

N is the set of natural numbers. Q is the field of complex rational numbers, i.e. the field of numbers a+bi, where a and b are rational. A sequence σ of elements of Q (i.e. a function σ, σ o:N→Q) is recursive iff there exist recursive functions fj(j=1,2,3,4; fj: N→N) such that for all n in N $$ \sigma (n) = \frac{{{f_1}(n)}} {{{f_2}\left( n \right) + 1}} + \frac{{{f_3}\left( n \right)}}{{{f_4}\left( n \right) + 1}}i$$ Such a sequence σ is recursively convergent iff there exists a recursive function k(k: N→N) such that for all h,j,n in N $$h \geqslant k(n)\Lambda \,j \geqslant k(n) \Rightarrow \left| {\sigma (h) - \sigma (j)} \right| < \frac{1} {{n + 1}}$$ (1)

CHAPTER TEN - ROOTS OF POLYNOMIALS

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

The original Novikov conjecture concerns the (oriented) homotopy invariance of higher sig-natures of manifolds and is equivalent to the rational injectivity of the assembly map in surgery theory. The integral injectivity of the assembly map is important for other purposes and is called the integral Novikov conjecture. There are also assembly maps in other theories and hence re-lated Novikov and integral Novikov conjectures. In this paper, we discuss several results on the integral Novikov conjectures for all torsion free arithmetic subgroups of linear algebraic groups and all S-arithmetic subgroups of reductive linear algebraic groups over number fields. For re-ductive linear algebraic groups over function fields of rank 0, the integral Novikov conjecture also holds for all torsion-free S-arithmetic subgroups. Since groups containing torsion elements occur naturally and frequently, we also discuss a generalized integral Novikov conjecture for groups containing torsion elements, and prove it for all arithmetic subgroups of reductive linear algebraic groups over number fields and S-arithmetic subgroups of reductive algebraic groups of rank 0 over function fields. 1

Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.