Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups

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dc.contributor.author Gromov, Mikhail
dc.date.accessioned 2015-10-08T07:38:17Z
dc.date.available 2015-10-08T07:38:17Z
dc.date.issued 1996
dc.identifier.citation Gromov, Mikhaıl. "Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups." Bull. Amer. Math. Soc 33 (1996): 0273-0979. en_US
dc.identifier.uri http://erepo.usiu.ac.ke/11732/1306
dc.description.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. en_US
dc.language.iso en en_US
dc.title Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups en_US
dc.type Article en_US

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