Expanders, Rank and Graphs of Groups
نویسنده
چکیده
A central principle of this paper is that, for a finitely presented group G, the algebraic properties of its finite index subgroups should be reflected by the geometry of its finite quotients. These quotients can indeed be viewed as geometric objects, in the following way. If we pick a finite set of generators for G, these map to a generating set for any finite quotient and hence endow this quotient with a word metric. This metric of course depends on the choice of generators, but if we were to pick another set of generators for G, the metrics on the quotients would change by a bounded factor. Thus, although the metric on any given finite quotient is unlikely to be useful, the metrics on the whole collection of finite quotients have a good deal of significance.
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