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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].
##### 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