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

- [Show abstract] [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 - [Show abstract] [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; - [Show abstract] [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

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.