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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 .
For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups. This terminology is also used when P is just another group.
The group G is a 2-group, that is, every element in G has finite order that is a power of 2. [1] The group G is periodic (as a 2-group) and not locally finite (as it is finitely generated). As such, it is a counterexample to the Burnside problem. The group G has intermediate growth. [2] The group G is amenable but not elementary amenable. [2]
The Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology named after Patrick H. Leslie. [1] [2] The Leslie matrix (also called the Leslie model) is one of the most well-known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration, growing in an unlimited ...
In the long run, exponential growth of any kind will overtake linear growth of any kind (that is the basis of the Malthusian catastrophe) as well as any polynomial growth, that is, for all α: = There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run).
Another definition of the Galois group comes from the Galois group of a polynomial []. If there is a field K / F {\displaystyle K/F} such that f {\displaystyle f} factors as a product of linear polynomials
He applied it to understand the asymptotic geometry of the word metric of a group of polynomial growth, by taking the limit of well-chosen rescalings of the metric. By tracking the limits of isometries of the word metric, he was able to show that the limiting metric space has unexpected continuities, and in particular that its isometry group is ...