A Note on Local Rigidity

The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way.     

Proximality and Equidistribution on the Furstenberg Boundary

Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and Γ a lattice in G. We prove that every Γ-orbits in the Furstenberg boundary G/P is equidistributed for the averages over Riemannian balls. The proof is based on the proximality of the action of Γ on G/P.*Partially supported by NSF grant 0400631. Mathematics Subject Classification (2000). 37B05, 22E40     

JSJ-Decompositions of finitely presented groups and complexes of groups

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger’s theory of group actions on CAT(0) spaces. Submitted: October 2003 Revision: February 2005 Accepted: June 2005     

Invariant Minimal Unit Vector Fields on Lie Groups

We provide a new characterization of invariant minimal unit vector fields on Lie groups and use it to construct some new examples. In particular, we determine all these vector fields on three-dimensional Lie groups.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

有限Chevalley群作用下的子代数轨道生成的格

讨论了一般情形Chevalley群作用下的子代数轨道生成的格.在同类型格中,研究了不同格之间的包含关系,并对格中子代数的特性及格的几何性进行了刻画.     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product. Received: January 2006, Revision: May 2006, Accepted: May 2006     

Extrapolated Magnus Methods

This paper describes the use of extrapolation with Magnus methods for the solution of a system of linear differential equations. The idea is a generalization of extrapolation with symmetric methods for the numerical solution of ODEs, where each extrapolation step increases the order of the method by 2.This revised version was published online in October 2005 with corrections to the Cover Date.     

Interaction of Flows on Quadrics

We interpret the Rodrigues formula describing a torsion of a rigid body in terms of Lie derivatives. We consider a more general situation where vector fields on quadrics (ellipsoids and hyperboloids) are dragged by other vector fields. The question is about the adjoint representation of a Lie group which is more general than the usual torsion group of the threedimensional Euclidean space.     

On the McCool group M3 and its associated Lie algebra

We prove that the Lie algebra of the McCool group M₃ is torsion free. As a result, we are able to give a presentation for the Lie algebra of M₃. Furthermore, M₃ is a Magnus group.