Asymptotic invariant of a group

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A property of a finitely-generated group $ G $ which is a quasi-isometry invariant of the metric space $ ( G,d _ {A} ) $, where $ d _ {A} $ is the word metric associated to a finite generating set $ A $ of $ G $( cf. also Quasi-isometric spaces). This definition does not depend on the choice of the set $ A $, since if $ B $ is another finite set of generators of $ G $, then the metric spaces $ ( G,d _ {A} ) $ and $ ( G,d _ {B} ) $ are quasi-isometric.

The theory of asymptotic invariants of finitely-generated groups has been recently brought to the foreground by M. Gromov (see, in particular, [a2] and [a3]). As Gromov says in [a3], p. 8, "one believes nowadays that the most essential invariants of an infinite group are asymptotic invariants" . For example, amenability (cf. Invariant average), hyperbolicity (in the sense of Gromov, cf. Hyperbolic group), the fact of being finitely presented (cf. Finitely-presented group), and the number of ends (cf. also Absolute) are all asymptotic invariants of finitely-generated groups. It is presently (1996) unknown whether the Kazhdan $ T $- property is an asymptotic invariant. For an excellent survey on these matters, see [a1].

A few examples of algebraic properties which are asymptotic invariants of finitely-generated groups are: being virtually nilpotent, being virtually Abelian, being virtually free.


[a1] E. Ghys, "Les groupes hyperboliques" Astérisque , 189–190 (1990) pp. 203–238 (Sém. Bourbaki Exp. 722)
[a2] M. Gromov, "Hyperbolic groups" S.M. Gersten (ed.) , Essays in Group Theory , MSRI Publ. , 8 , Springer (1987) pp. 75–263
[a3] M. Gromov, "Asymptotic invariants of infinite groups" , Proc. Symp. Sussex, 1991: II , London Math. Soc. Lecture Notes , 182 , Cambridge Univ. Press (1993) pp. 1–291
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This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article