Presentations of the first homotopy groups of the unitary groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n 2 we supply n homotopic maps that each represent the (n - 1)!-th power of a suitable generator of _2n SU(n) _n!. The product of these n commuting maps is the constant map to the identity matrix.     

PF环与群环的Grothendieck群

Let R be a commutative ring with 1, G an Abelian group, RG the group ring on R and G. In this paper we gave some properties of PF- rings in which f. g. projective modules are free. The Grothendieck groups K₀(RG) for some cases are given. In addition, for the ring R with the unimodular column property, we proved the following result: K₀(RG) ≈K₀(R), hence if R ∈PF, then K₀(RG)≈Z .     

Embedding of baumslag-solitar groups into the generalized Baumslag-Solitar groups

A finitely generated group G that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag-Solitar group or GBS-group. Let p and q be coprime integers other than 0, 1, and ?1. We prove that the Baumslag-Solitar group BS(p, q) embeds into G if and only if the equation x ^?1 y ^p x = y ^q is solvable in G for y ≠ 1; i.e., $\tfrac{p} {q} $ ∈ Δ(G), where Δ is the modular homomorphism.     

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.     

On the groups of unitriangular automorphisms of relatively free groups

We describe the structure of the group U _n of unitriangular automorphisms of the relatively free group G _n of finite rank n in an arbitrary variety C of groups. This enables us to introduce an effective concept of normal form for the elements and present U _n by using generators and defining relations. The cases n = 1, 2 are obvious: U ₁ is trivial, and U ₂ is cyclic. For n ?? 3 we prove the following: If G _n?1 is a nilpotent group then so is U _n . If G _n?1 is a nilpotent-by-finite group then U _n admits a faithful matrix representation. But if the variety C is different from the variety of all groups and G _n?1 is not nilpotent-by-finite then U _n admits no faithful matrix representation over any field. Thus, we exhaustively classify linearity for the groups of unitriangular automorphisms of finite rank relatively free groups in proper varieties of groups, which complements the results of Olshanskii on the linearity of the full automorphism groups AutG _n . Moreover, we introduce the concept of length of an automorphism of an arbitrary relatively free group G _n and estimate the length of the inverse automorphism in the case that it is unitriangular.     

Surgery groups of the fundamental groups of hyperplane arrangement complements

Using a recent result of Bartels and Lück (The Borel conjecture for hyperbolic and CAT(0)-groups (preprint) \({{\tt arXiv:0901.0442v1}}\)) we deduce that the Farrell–Jones Fibered Isomorphism conjecture in \({L^{\langle -\infty \rangle}}\)-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Artin full braid groups. As a consequence we explicitly compute the surgery groups of the Artin pure braid groups. This is obtained as a corollary to a computation of the surgery groups of a more general class of groups, namely for the fundamental group of the complement of any fiber-type hyperplane arrangement in \({{\mathbb C}^n}\).     

Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

On the reconstruction index of permutation groups: semiregular groups

Summary. The reconstruction index of all semiregular permutation groups is determined. We show that this index satisfies 3 ￡ r(G, W) ￡ 5 3 \leq \rho(G, \Omega) \leq 5 and we classify the groups in each case.