Boundedly generated groups and small-cancellation method

4 March 2008
Alex Muranov
A group is called boundedly generated if it is the product of a finite sequence of its cyclic subgroups. Bounded generation is a property possessed by finitely generated abelian groups and by some other linear groups. Apparently it was not known before whether all boundedly generated groups are linear. Another question about such groups has also been open for a while: If a torsion-free group $G$ has a finite sequence of generators $a_1,\dotsc,a_n$ such that every element of $G$ can be written in a unique way as $a_1^{k_1}\dotsm a_n^{k_n}$, where $k_i\in\mathbb Z$, is it true then that $G$ is virtually polycyclic? (Vasiliy Bludov, Kourovka Notebook, 1995.) Counterexamples to resolve these two questions have been constructed using small-cancellation method of combinatorial group theory. In particular boundedly generated simple groups have been constructed.