Invariant sets and measures of nonexpansive group automorphisms

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS ³. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.     

A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I

Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of  explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.

     

A product formula for spherical representations of a group of automorphisms of a homogeneous tree, II

Let be the group of automorphisms of a homogeneous tree and let be the tensor product of two spherical irreducible unitary representations of . We complete the explicit decomposition of commenced in part I of this paper, by describing the discrete series representations of which appear as subrepresentations of .

     

Fixed point properties for nonexpansive representations of topological semigroups

In this paper, we study a fixed point and a nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings on a nonempty subset of a Hilbert space.     

Local group representations

As a generalization of the usual notion of a group representation, a local group representation of connected and locally connected groups G is introduced as a certain sheaf A with base G, such that all fibres A _x may be identified with a fixed linear subspace D of the representation space. The sheaf provides a means of expressing a group symmetry which is known to hold only locally (in a sense which will be made precise). The problem of extending a local representation to a global one may be considered as a noncommutative counterpart of the dilation theory of systems of doubly commuting contractions. If G is a Lie group, one may associate a derived representation of the enveloping algebra (g) of the Lie algebra g of G. In case of isometric local representations in a Hilbert space, these are *-representations and determine the appropriate local representation uniquely. As an application we obtain that the n-tuples of antisymmetric operators of partial derivatives on L ²(), where is a connected open subset of ⁿ, determine up to translation.     

Ergodic group automorphisms are exponentially recurrent

We show that ergodic automorphisms of compact abelian groups have the property that for every nonempty open setU, the measure of the set first returning toU aftern iterates decays exponentially inn. This follows from a result about aperiodic automorphisms of countable abelian groups, whose proof employsp-adic analysis. Partially supported by NSF Grant MCS 7704915.     

Functions realizing as abelian group automorphisms

Let A be a set and f:A→A a bijective function. Necessary and sufficient conditions on f are determined which makes it possible to endow A with a binary operation ? such that (A,?) is a cyclic group and f∈Aut(A). This result is extended to all abelian groups in case |A| = p², p a prime. Finally, in case A is countably infinite, those f for which it is possible to turn A into a group (A,?) isomorphic to ?ⁿ for some n≥1, and with f∈Aut(A), are completely characterized.     

On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

Hall's universal group has ample generic automorphisms

We show that the automorphism group of Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Consequently, it has the small index property, is not the union of a countable chain of non‐open subgroups, and has the automatic continuity property. Also, we discuss some algebraic and topological properties of the automorphism group of Hall's universal group. For example, we show that every generic automorphism of Hall's universal group is conjugate to all of its powers, and hence has roots of all orders.     

The centralizer of a set of group automorphisms

Let A be a set of automorphisms of a finite group G. The structure of the near-ring C(A) of identity preserving functions which commute with every element of A is investigated.