Let V be a pseudovariety of finite groups such that free groups are residually V, and let φ:F (A) → F (B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ̂ of φ between the pro-V completions of F (A) and F (B) is also injective. In particular, if V is extension-closed, this is the case if and only if φ(F (A)) and its...

The Tango bundle T over P is proved to be the pull-back of the twisted Cayley bundle C(1) via a map f : P → Q5 existing only in characteristic 2. The Frobenius morphism φ factorizes via such f . Using f the cohomology of T is computed in terms of S ⊗C, φ∗(C), Sym(C) and C, while these are computed by applying Borel-Bott-Weil theorem. By machine-aided computation the mimimal resolutions of C and...

Let V be a G-module where G is a complex reductive group. Let Z := V//G denote the categorical quotient and let π : V → Z be the morphism dual to the inclusion O(V ) ⊂ O(V ). Let φ : Z → Z be an algebraic automorphism. Then one can ask if there is an algebraic map Φ: V → V which lifts φ, i.e., π(Φ(v)) = φ(π(v)) for all v ∈ V . In Kuttler [Kut11] the case is treated where V = rg is a multiple of...

We introduce the notion of a hyper-atom. One of the main results of this paper is the 2|G| 3 –Theorem: Let S be a finite generating subset of an abelian group G of order ≥ 2. Let T be a finite subset of G such that 2 ≤ |S| ≤ |T |, S + T is aperiodic, 0 ∈ S ∩ T and 2|G|+ 2 3 ≥ |S + T | = |S|+ |T | − 1. Let H be a hyper-atom of S. Then S and T are H–quasi-periodic. Moreover φ(S) and φ(T ) are ari...

Fixed points u = φ(u) of marked and primitive morphisms φ over arbitrary alphabet are considered. We show that if u is palindromic, i.e., its language contains infinitely many palindromes, then some power of φ has a conjugate in class P. This class was introduced by Hof, Knill and Simon (1995) in order to study palindromic morphic words. Our definitions of marked and well-marked morphisms are m...

The Hall topology for the free group is the coarsest topology such that every group morphism from the free group onto a finite discrete group is continuous. It was shown by M. Hall Jr that every finitely generated subgroup of the free group is closed for this topology. We conjecture that if H1, H2, . . . , Hn are finitely generated subgroups of the free group, then the product H1H2 · · · Hn is ...

Given a generically surjective holomorphic vector bundle morphism f : E → Q, E and Q Hermitian bundles, we construct a current R with values in Hom (Q,H), where H is a certain derived bundle, and with support on the set Z where f is not surjective. The main property is that if φ is a holomorphic section of Q, and Rφ = 0, then locally fψ = φ has a holomorphic solution ψ. In the generic case also...

We consider all complex projective manifolds X that satisfy at least one of the following three conditions: (1) There exists a pair (C , φ), where C is a compact connected Riemann surface and φ : C −→ X a holomorphic map, such that the pull back φ∗TX is not semistable. (2) The variety X admits an étale covering by an abelian variety. (3) The dimension dimX ≤ 1. We prove that the following class...

Let (X,x) be a germ of an isolated canonical 3-fold singularity. Fix a representative X of the germ which is Stein and contractible. Let φ′ : Y ′ → X be a crepant projective morphism from Y ′ with terminal Q-factorial singularities, and ψ : Y → Y ′ an analytic Q-factorialisation of Y ′ at its singular points. The aim of this paper is to calculate the Betti numbers of Y or, equivalently, the int...