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A property of a [[Finitely-generated group|finitely-generated group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107801.png" /> which is a [[Quasi-isometry|quasi-isometry]] invariant of the [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107802.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107803.png" /> is the [[Word metric|word metric]] associated to a finite generating set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107804.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107805.png" /> (cf. also [[Quasi-isometric spaces|Quasi-isometric spaces]]). This definition does not depend on the choice of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107806.png" />, since if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107807.png" /> is another finite set of generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107808.png" />, then the metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a1107809.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a11078010.png" /> are quasi-isometric.
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The theory of asymptotic invariants of finitely-generated groups has been recently brought to the foreground by M. Gromov (see, in particular, [[#References|[a2]]] and [[#References|[a3]]]). As Gromov says in [[#References|[a3]]], p. 8,  "one believes nowadays that the most essential invariants of an infinite group are asymptotic invariants" . For example, amenability (cf. [[Invariant average|Invariant average]]), hyperbolicity (in the sense of Gromov, cf. [[Hyperbolic group|Hyperbolic group]]), the fact of being finitely presented (cf. [[Finitely-presented group|Finitely-presented group]]), and the number of ends (cf. also [[Absolute|Absolute]]) are all asymptotic invariants of finitely-generated groups. It is presently (1996) unknown whether the Kazhdan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110780/a11078012.png" />-property is an asymptotic invariant. For an excellent survey on these matters, see [[#References|[a1]]].
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A property of a [[Finitely-generated group|finitely-generated group]]  $  G $
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which is a [[Quasi-isometry|quasi-isometry]] invariant of the [[Metric space|metric space]]  $  ( G,d _ {A} ) $,
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where  $  d _ {A} $
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is the [[Word metric|word metric]] associated to a finite generating set  $  A $
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of  $  G $(
 +
cf. also [[Quasi-isometric spaces|Quasi-isometric spaces]]). This definition does not depend on the choice of the set  $  A $,
 +
since if  $  B $
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is another finite set of generators of  $  G $,
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then the metric spaces  $  ( G,d _ {A} ) $
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and  $  ( G,d _ {B} ) $
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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, [[#References|[a2]]] and [[#References|[a3]]]). As Gromov says in [[#References|[a3]]], p. 8,  "one believes nowadays that the most essential invariants of an infinite group are asymptotic invariants" . For example, amenability (cf. [[Invariant average|Invariant average]]), hyperbolicity (in the sense of Gromov, cf. [[Hyperbolic group|Hyperbolic group]]), the fact of being finitely presented (cf. [[Finitely-presented group|Finitely-presented group]]), and the number of ends (cf. also [[Absolute|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 [[#References|[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.
 
A few examples of algebraic properties which are asymptotic invariants of finitely-generated groups are: being virtually nilpotent, being virtually Abelian, being virtually free.

Revision as of 18:48, 5 April 2020


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.

References

[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
How to Cite This Entry:
Asymptotic invariant of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_invariant_of_a_group&oldid=11220
This article was adapted from an original article by A. Papadopoulos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article