Article

# On the cardinality of sumsets in torsion-free groups

Bulletin of the London Mathematical Society (Impact Factor: 0.7). 09/2010; DOI: 10.1112/blms/bds032
Source: arXiv

ABSTRACT Let $A, B$ be finite subsets of a torsion-free group $G$. We prove that for every positive integer $k$ there is a $c(k)$ such that if $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0k^{6}$ for arbitrary torsion-free groups, and $c(k)<c_0k^{3}$ for groups with the unique product property, where $c_0$ is an absolute constant. We give examples to show that $c(k)$ is at least quadratic in $k$.

0 Bookmarks
·
83 Views
• Source
##### Article: A Structure Theorem for Small Sumsets in Nonabelian Groups
[Hide abstract]
ABSTRACT: Let G be an arbitrary finite group and let S and T be two subsets such that |S|>1, |T|>1, and |TS|< |T|+|S|< |G|-1. We show that if |S|< |G|-4|G|^{1/2}+1 then either S is a geometric progression or there exists a non-trivial subgroup H such that either |HS|< |S|+|H| or |SH| < |S|+|H|. This extends to the nonabelian case classical results for Abelian groups. When we remove the hypothesis |S|<|G|-4|G|^{1/2}+1 we show the existence of counterexamples to the above characterization whose structure is described precisely.
European Journal of Combinatorics 03/2012; · 0.61 Impact Factor
• Source
##### Article: On geometric aspects of diffuse groups
[Hide abstract]
ABSTRACT: Bowditch introduced the notion of diffuse groups as a geometric variation of the unique product property. We elaborate on various examples and non-examples, keeping the geometric point of view from Bowditch's paper. In particular, we discuss fundamental groups of flat and hyperbolic manifolds. The appendix settles an open question by providing an example of a group which is diffuse but not left-orderable.
11/2014;
• Source
##### Article: New Examples of Torsion-Free Non-unique Product Groups
[Hide abstract]
ABSTRACT: We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.
Journal of Group Theory 01/2013; 3(3). · 0.35 Impact Factor