A Bieberbach Theorem for crystallographic group extensions

Cauchy–davenport Theorem in Group Extensions

Let A and B be nonempty subsets of a finite group G in which the order of the smallest nonzero subgroup is not smaller than d = |A| + |B| − 1. Then at least d different elements of G has a representation in the form ab, where a ∈ A and b ∈ B. This extends a classical theorem of Cauchy and Davenport to noncommutative groups. We also generalize Vosper’s inverse theorem in the same spirit, giving ...

Group actions on one-manifolds, II: Extensions of Hölder’s Theorem

This self-contained paper is part of a series [FF1] seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on R with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in Diff(R) as those groups whose elements have...

The Bieberbach Case in Gromov's Almost Flat Manfold Theorem

Extensions of Perov theorem

[Perov, A. I., On Cauchy problem for a system of ordinary diferential equations, (in Russian), Priblizhen. Metody Reshen. Difer. Uravn., 2 (1964), 115-134] used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article we study fixed point results for the new extensions of Banach’s contraction principle ...

Brégman’s theorem and extensions

Minc conjectured, and Brégman proved, a sharp upper bound on the permanent of an n by n 0-1 matrix with given row sums (equivalently, on the number of perfect matchings in a bipartite graph with each partition class having size n and with fixed degree sequence for one of the two classes). Here we present Radhakrishnan’s entropy proof of Brégman’s theorem, and Alon and Friedland’s proof of an an...