Strong approximation methods in group theory, an LMS/EPSRC Short course lecture notes

These are the lecture notes for the LMS/EPSRC short course on strong approximation methods in linear groups organized by Dan Segal in Oxford in September 2007.Comment: v4: Corollary 6.2 corrected, added a few small remark

Strange images of profinite groups

We investigate whether a finitely generated profinite group G could have a finitely generated infinite image. A result of Dan Segal shows that this is impossible if G is prosoluble. We prove that such an image does not exist if G is semisimple or nonuniversal. We also investigate the existense of dense normal subgroups in $G$.Comment: The results of this preprint have been superceded by http://arxiv.org/abs/1102.3037 which answers the questions posed her

Algebraic properties of profinite groups

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update