Showing papers on "Arithmetic group published in 2001"
TL;DR: In this paper, the existence of a smooth projective curve over F 2 and representations of the arithmetic fundamental group of X⊗k with values in SL 2 (k[[t]]), with k suitable finite field of characteristic 2, such that the image of the geometric fundamental group is infinite was shown.
Abstract: One shows the existence of a smooth projective curve over F 2 and of representations of the arithmetic fundamental group of X⊗k with values in SL 2 (k[[t]]) , with k suitable finite field of characteristic 2, such that the image of the geometric fundamental group is infinite. This gives a negative answer to a question of A.J. de Jong.
19 citations
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TL;DR: In particular, it avoids recourse to the classification theorems as mentioned in this paper, and thus avoids the need for a formal classification of the arithmetic groups in the form of a set of automorphisms.
Abstract: Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again arithmetic. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems. It was originally completed on March 28, 1984, and distributed in handwritten form. July 23, 2001: Fixed about 30 misprints.
9 citations
1 Jan 2001
TL;DR: In this article, the authors present counter-examples to the unstable version of a conjecture postulating the relationship between cohomology of arithmetic groups and the "etale models" of their classifying spaces.
Abstract: In this paper, we present counter-examples to the unstable version of a conjecture postulating the relationship between cohomology of arithmetic groups and the “etale models” of their classifying spaces.
5 citations
TL;DR: In this article, it was shown that the isomorphism type of the Q-algebra Q[G] determines the commensurability class of U(Z[G]) of the integral group ring Z[G], subject to a certain restriction on the Q representations of G.
Abstract: For any finite group G the group U(Z[G]) of units in the integral group ring Z[G]is an arithmetic group in a reductive algebraic group, namely the Zariski closure of SL1 (Q[G]). In particular, the isomorphism type of the Q-algebra Q[G] determines the commensurability class of U(Z[G]); we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the Q-representations of G the converse is exactly true.
2 citations
1 Jan 2001
TL;DR: In this paper, it was shown that the universal arithmetic group U0 is a cocompact subgroup U of PSL(2,C) which has the property that every closed, oriented 3-manifold M is homeomorphic to the quotient H3/G for some finite-index subgroup G of U. In other words, M is the underlying space of some finite orbifold cover of H 3/U.
Abstract: Let H3 denote hyperbolic 3-space and identify its group of orientation-preserving isometries with PSL(2,C). In the context of usage here, a universal group is a cocompact subgroup U of PSL(2,C) which has the property that every closed, oriented 3-manifold M is homeomorphic to the quotient H3/G for some finite-index subgroup G of U. In other words, every closed, oriented 3-manifold M is the underlying space of some finite orbifold cover of H3/U. The authors of this paper, jointly with W. C. Whitten, previously constructed a particular group U0 which they showed to be universal [Invent. Math. 87 (1987), no. 3, 441–456;]. In this work, they demonstrate that U0 is in fact arithmetic. Specifically, let F denote the unique quartic field of discriminant −400 (which has one complex place) and let A denote the quaternion algebra over F which is ramified only at the two real places. They show that A contains a particular order whose group of elements of norm one is commensurable with U0. It is suggested that the arithmetic structure may be of use in studying this group (in particular, the finite index subgroups of U0 generated by elliptic elements are relevant to the Poincare conjecture). They further conjecture that this is the simplest such example: more precisely, they conjecture that 400 is the smallest absolute value of the invariant trace field discriminant of any universal arithmetic group.
2 citations
TL;DR: In this article, it was shown that the S -arithmetic group SO n (f) O (S) has bounded generation, where f is a nondegenerate quadratic form in n⩾5 variables and S is a finite set of places of K containing all archimedean places.
Abstract: Let f be a nondegenerate quadratic form in n⩾5 variables and of Witt index ⩾2 over a number field K , S be a finite set of places of K containing all archimedean places. We prove that the S -arithmetic group SO n (f) O (S) has bounded generation.