Suppose that G is a locally compact group and U is a representation of a closed subgroup H of G on a Hilbert space H. We use nonabelian duality for crossed products of C∗-algebras to study the following problem: when does U extend to a representation of G on the same space H?

We use presentations and identities among relations to give a generalisation of the Schreier theory of nonabelian extensions of groups. This replaces the usual multiplication table for the extension group by more efficient, and often geometric, data. The methods utilise crossed modules and crossed resolutions.

Given a one-relator group, we give an effective estimate on the minimal index of a subgroup with a nonabelian free quotient. We show that the index is bounded by a polynomial in the length of the relator word. We also provide a lower bound on the index.

Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if G 1 , G 2 are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over G 1 is stably orbit equivalent to every nontrivial Bernoulli shift over G 2. This answers a question of S. Popa.

We study D-branes on a three complex dimensional nonabelian orbifold C/Γ with Γ a finite subgroup of SU(3). We present general formulae which are necessary to obtain the gauge group and the spectrum of the D-brane worldvolume theory for dihedral-like subgroups ∆(3n) and ∆(6n). Physical implications of the formulae are discussed. ∗ e-mail address: [email protected]

We analyze a nonabelian extension of Born–Infeld action for the SU(2) group. In the class of spherically symmetric solutions we find that, besides the Gal’tsov–Kerner glueballs, only the analytic dyons have finite energy. The presented analytic and numerical investigation excludes the existence of pure magnetic monopoles of ’t Hooft–Polyakov type.

We study two–generated subgroups x f , gy Homeo pIq such that x f 2, g2y is isomorphic to Thompson’s group F. We show that this class contains examples with nonabelian free subgroups, examples which do not admit faithful actions by C diffeomorphisms on 1–manifolds, and examples which are not finitely presented. We thus answer questions due to M. Brin.

In this paper a first step in classifying the fuzzy subgroups of a finite nonabelian group is made. We develop a general method to count the number of distinct fuzzy subgroups of such groups. Explicit formulas are obtained in the particular case of dihedral groups.

An explicit expression for the ε-factor εK((V,N),ψ,dμ) of a representation (V,N) of the Weil-Deligne group WDK of a local field K is given in terms of the nonabelian local class field theory of K.