### Abstract:

Geometric group theory, Vol. 2: Asymptotic invariants of inﬁnite groups,byM.
Gromov, edited by A. Niblo and Martin A. Roller, London Math. Soc. Lecture
Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, vii + 295 pp.,
$34.95, ISBN 0-521-44680-5
The graphic work of M. C. Escher is much appreciated by mathematicians and
is also enjoyed by a much wider public. A quick look at Escher’s pictures of tesselations
of the Euclidean and hyperbolic planes makes everyone see that these
two geometries have diﬀerent asymptotic behavior: the Euclidean plane has quadratic
polynomial growth, while the hyperbolic plane has exponential growth. The
reviewerhasfound, inteachingcoursestoaudiencesnotatallinclinedtomathematics,thatthediﬀerencebetweenthesetwogeometriescanbeimmediatelyunderstood
and appreciated by looking at these pictures.
The evident quadratic and exponential growth that can be seen in these pictures
is a special case of one of the earliest and most easily understandable asymptotic
invariant of inﬁnite (and ﬁnitely generated) groups, namely, the growth of a group.
Together with the concept of growth came the idea of considering a group to be a
metric space and to relate this metric space to more conventional metric spaces on
which the group acts by the concept of quasi-isometry. Namely, if Γ is a ﬁnitely
generated group and x ∈ Γ, deﬁne |x|, the distance of x from the identity, relative
to a set of generators x1,···,xn for Γ, to be the minimum length of a word in
x±1
1 ,···,x±1n expressing the element x ∈ Γ. The distance between x,y ∈ Γis
deﬁned to be |x−1y|. It is easily checked that this distance makes Γ into a metric
spaceandthatdiﬀerent(ﬁnite)setsofgeneratorsgivemetricswhicharebi-Lipschitz
equivalent. The growth function of Γ is the function on the natural numbers that
ton associates the cardinality of the ball of radius n, i.e., the number of elements in
Γ expressible as words of length at most n.Thegrowth of Γ is the order of growth
of this function.
The original applications of the idea of viewing a group as a metric space can
nowbe formalizedinthe conceptof quasi-isometry. TwometricspacesX andY are
called quasi-isometric if there exists a map.