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The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length at most n (relative to a symmetric generating set) is bounded above by a polynomial function p(n). The order of growth is then the least degree of any such polynomial ...
A finite group has constant growth—that is, polynomial growth of order 0—and this includes fundamental groups of manifolds whose universal cover is compact. If M is a closed negatively curved Riemannian manifold then its fundamental group π 1 ( M ) {\displaystyle \pi _{1}(M)} has exponential growth rate.
A linear group is not amenable if and only if it contains a non-abelian free group (thus the von Neumann conjecture, while not true in general, holds for linear groups). The Tits alternative is an important ingredient [2] in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result ...
See Gromov's theorem on groups of polynomial growth. (Also see D. Edwards for an earlier work.) (Also see D. Edwards for an earlier work.) The key ingredient in the proof was the observation that for the Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense.
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H ≤ G {\displaystyle H\leq G} such that H has property P.
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Finite groups — Group representations are a very important tool in the study of finite groups.
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The Breuillard–Green–Tao theorem on classification of approximate groups can be used to give a new proof of Gromov's theorem on groups of polynomial growth.The result obtained is actually a bit stronger since it establishes that there exists a "growth gap" between virtually nilpotent groups (of polynomial growth) and other groups; that is, there exists a (superpolynomial) function such ...