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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.identifier.uri | ||

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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