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A property of a [[Finitely-generated group|finitely-generated group]]  $  G $
which is a [[Quasi-isometry|quasi-isometry]] invariant of the [[Metric space|metric space]]  $  ( G,d _ {A} ) $, 
where  $  d _ {A} $
is the [[Word metric|word metric]] associated to a finite generating set  $  A $
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 $
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, [[#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.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Ghys,   "Les groupes hyperboliques"  ''Astérisque'' , '''189–190'''  (1990)  pp. 203–238  (Sém. Bourbaki Exp. 722) {{ZBL|0744.20036}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Gromov,   "Hyperbolic groups"  S.M. Gersten (ed.) , ''Essays in Group Theory'' , ''MSRI Publ.'' , '''8''' , Springer  (1987)  pp. 75–263</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR>
</table>