Group-theoretic applications of non-commutative toric geometry.

*(English)*Zbl 0935.20028
Campbell, C. M. (ed.) et al., Groups St. Andrews 1997 in Bath. Selected papers of the international conference, Bath, UK, July 26-August 9, 1997. Vol. 1. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 260, 176-194 (1999).

This paper consists of lecture notes in which the author considers geometric techniques applied to questions about “small” groups, that is, groups containing no non-Abelian free subgroups. The author studies in particular their finite presentability and the structure of their automorphism groups. The context is most of the time that of discrete groups, but towards the end of the paper, there is a discussion on pro-\(p\) groups.

The common theme is that a small group is in some sense at most half the size of a free group. The author looks at various aspects of the representation theory of the groups involved. The author looks at the residually finite images of the groups. The techniques involve studying filtrations of group rings. The author shows that some non-commutative versions of familiar Noetherian rings are useful in that their representation theory has applications in group theory. The filtrations that are used are connected to a particular type of group actions on trees, and group automorphisms are studied via a construction related to the compactification of complex varieties.

For the entire collection see [Zbl 0908.00017].

The common theme is that a small group is in some sense at most half the size of a free group. The author looks at various aspects of the representation theory of the groups involved. The author looks at the residually finite images of the groups. The techniques involve studying filtrations of group rings. The author shows that some non-commutative versions of familiar Noetherian rings are useful in that their representation theory has applications in group theory. The filtrations that are used are connected to a particular type of group actions on trees, and group automorphisms are studied via a construction related to the compactification of complex varieties.

For the entire collection see [Zbl 0908.00017].

Reviewer: A.Papadopoulos (Strasbourg)

##### MSC:

20F65 | Geometric group theory |

20F05 | Generators, relations, and presentations of groups |

13E05 | Commutative Noetherian rings and modules |

20E07 | Subgroup theorems; subgroup growth |

20F28 | Automorphism groups of groups |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

16S34 | Group rings |

16W70 | Filtered associative rings; filtrational and graded techniques |

20E08 | Groups acting on trees |