Gradual transitivity in orthogonality spaces of finite rank

An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements e and f, we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps e to f, and any of the automorphisms leaves the elements orthogonal to e and f fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.

FINITENESS OF TOPOLOGICAL ENTROPY FOR LOCALLY COMPACT ABELIAN GROUPS

We study the locally compact abelian groups in the class ${\mathfrak E_{ \lt \infty }}$ , that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$ , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to ${\mathfrak E_{ \lt \infty }}$ , and that those of them that belong to $\mathfrak E_0$ are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.

SEMIABELIAN VARIETIES OVER SEPARABLY CLOSED FIELDS, MAXIMAL DIVISIBLE SUBGROUPS, AND EXACT SEQUENCES

Given a separably closed field $K$ of characteristic $p>0$ and finite degree of imperfection, we study the $\sharp$ functor which takes a semiabelian variety $G$ over $K$ to the maximal divisible subgroup of $G(K)$. Our main result is an example where $G^{\sharp }$, as a ‘type-definable group’ in $K$, does not have ‘relative Morley rank’, yielding a counterexample to a claim in Hrushovski [J. Amer. Math. Soc. 9 (1996), 667–690]. Our methods involve studying the question of the preservation of exact sequences by the $\sharp$ functor, and relating this to issues of descent as well as model-theoretic properties of $G^{\sharp }$. We mention some characteristic 0 analogues of these ‘exactness-descent’ results, where differential algebraic methods are more prominent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, which is interesting in its own right, as well as providing a uniform treatment of the characteristic 0 and characteristic $p$ cases of ‘exactness descent’.

Weil–Châtelet divisible elements in Tate–Shafarevich groups I: The Bashmakov problem for elliptic curves over

For an abelian variety $A$ over a number field $k$ we discuss the maximal divisible subgroup of ${\mathrm{H} }^{1} (k, A)$ and its intersection with the subgroup Ш$(A/ k)$. The results are most complete for elliptic curves over $ \mathbb{Q} $.

Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

Schur-Zassenhaus theorem revisited

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.Theorem 2. Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.This result has been proved in [1] with the additional assumption that G is connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.The second author has shown in [11] that a nilpotent ω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus.

Note on the extensions of Butler groups

In terms of a new notion, called prebalancedness, those extensions of a Butler group by a Butler group which are again Butler groups are characterised. It is shown that the prebalanced extensions of torsion-free abelian groups form a divisible subgroup in the group of extensions.