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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 ...
The free abelian group has a polynomial growth rate of order d. The discrete Heisenberg group H 3 {\displaystyle H_{3}} has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Hyman Bass and Yves Guivarch that is discussed in the article on Gromov's theorem .
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.
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 ...
Gromov's theorem on groups of polynomial growth; Growth rate (group theory) H. ... (mathematics) Residually finite group; Residue-class-wise affine group; S.
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods ...
Beyond theoretical computer science, quasi-polynomial growth bounds have also been used in mathematics, for instance in partial results on the Hirsch conjecture for the diameter of polytopes in polyhedral combinatorics, [8] or relating the sizes of cliques and independent sets in certain classes of graphs. [9]