Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11]. Received: 21 April 2005     

Central extensions of gauge groups revisited

We present an explicit construction for the central extension of the group Map(X, G) where X is a compact manifold and G is a Lie group. If X is a complex curve we obtain a simple construction of the extension by the Picard variety Pic(X). The construction is easily adapted to the extension of Aut(E), the gauge group of automorphisms of a nontrivial vector bundle E.     

混沌群作用

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Approximations of stable actions on $ \Bbb {R} $-trees

This article shows how to approximate a stable action of a finitely presented group on an -tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an -tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on -trees and on the dynamical study of exotic components by D. Gaboriau. Received: 22 October, 1997     

Quasi-isometries between groups with infinitely many ends

Let G, F be finitely generated groups with infinitely many ends and let? be graph of groups decompositions of F, G such that all edge groups are finite and all vertex groups have at most one end. We show that G, F are quasi-isometric if and only if every one-ended vertex group of is quasi-isometric to some one-ended vertex group of and every one-ended vertex group of is quasi-isometric to some one-ended vertex group of?. From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: and are all quasi-isometric. Received: April 7, 2000; revised version: October 6, 2000     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

Geometry for palindromic automorphism groups of free groups

We examine the palindromic automorphism group , of a free group F _n , a group first defined by Collins in [5] which is related to hyperelliptic involutions of mapping class groups, congruence subgroups of , and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible space on which acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about the cohomology of , such as that its rational cohomology is zero at the vcd. Received: January 17, 2000.     

Tensor products of singular representations and an extension of the $ \theta $-correspondence

In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space . Our procedure establishes a correspondence between certain unitary representations of G and those of . This extends the usual -correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E ₇ and F ₄.     

The decomposition of 3-dimensional Poincaré complexes

We show that if the fundamental group of an orientable PD ³-complex has infinitely many ends then it is either a proper free product or virtually free of finite rank. It follows that every PD ³-complex is finitely covered by one which is homotopy equivalent to a connected sum of aspherical PD ³-complexes and copies of . Furthermore, it is shown that any torsion element of the fundamental group of an orientable PD ³-complex has finite centraliser. Received: February 2, 1998     

Projective normality of G.I.T. quotient varieties modulo finite groups

We prove that for any finite dimensional representation V of a finite group G of order n the quotient variety G??(V) is projectively normal with respect to descent of 𝒪(1)^?l where l = lcm{1,2,3,4,…,n}. We also prove that for the tautological representation V of the alternating group A_n the projective variety A_n??(V) is projectively normal with respect to the descent of the above line bundle.     

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let be a quotient of X by a torsion-free discrete subgroup of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual X _c , the locally symmetric structure of M can be considered as a special kind of a -structure on M, a maximal atlas of X _c -valued charts with locally constant transition maps in the complexified group . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung. Received: September 23, 1999     

Rotationally Invariant Rank 1 Convex Functions

Let f be a function on the set M ⁿ xn of all n by n real matrices. If f is rotationally invariant with respect to the proper orthogonal group, it has a representation \tilde f through the signed singular values of the matrix argument ?∈ M^nxn. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of \tilde f . Accepted 20 December 2000. Online Publication 18 May, 2001.     

The topology at infinity of Coxeter groups and buildings

The connectivity at infinity of a finitely generated Coxeter group W is completely determined by topological properties of its nerve L (a finite simplicial complex). For example, W is simply connected at infinity if and only if L and the subcomplexes (where ranges over all simplices in L) are simply connected. This characterization extends to locally finite buildings. Received: May 3, 2001     

Essential cohomology of finite groups

In this paper we prove that a finite group G with Cohen-Macaulay mod p cohomology will have non-trivial undetectable elements in if and only if G is a p-group such that every element of order p in G is central. Applications and examples are also provided. Received: April 18, 1996     

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character represents a point in the homotopical or homological -invariants of the group G. Let be a finite simplicial graph, the flag complex induced by , and the graph group, or 'right angled Artin group', defined by . We use our result on n-generating subgroup sets to describe the homotopical and homological -invariants of in terms of the topology of subcomplexes of . In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the -invariants for a family of groups whose higher invariants are not determined - either implicitly or explicitly - by ¹. Received: October 18, 1996     

Local constancy in families of non-abelian Galois representations

If is a a scheme of finite type over a local field F, and is a proper smooth family, then to each rational point one can assign an extension of the absolute Galois group of F by the geometric fundamental group G of the fibre . If F has uniformiser , and residue characteristic p, we show that the corresponding extension of the absolute Galois group of by the maximal prime to p quotient of G is locally constant in the -adic topology of . We give a similar result in the case of non-proper families, and families over -adic analytic spaces. Received August 14, 1998     

The conjugacy problem for Dehn twist automorphisms of free groups

A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn twist automorphisms of . Received: February 12, 1996.     

Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions

To any compact hyperbolic Riemann surface X, we associate a new type of automorphism group — called its commensurability automorphism group, ComAut(X). The members of ComAut(X) arise from closed circuits, starting and ending at X, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by X (in $ T_\infty $), for the action of the universal commensurability modular group on the universal direct limit of Teichmüller spaces, $ T_\infty $. Now, each point of $ T_\infty $ represents a complex structure on the universal hyperbolic solenoid. We notice that ComAut(X) acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing X is arithmetic. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over $ T_\infty $, are studied by us.     

The primitive permutation groups of degree 2 a ·3 b

In this paper, we give a complete classification of all finite simple groups with maximal subgroups of index n, where n = 2^a·3^b for a, b≧ 1. As a consequence, for such n, all primitive permutation groups of degree n are given. The motivation of this work comes also from a study of Cayley graphs of certain valency on a finite simple group. Received: 9 March 2005