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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.
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 the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79–80 in [1]).
Subgroup growth studies these functions, their interplay, and the characterization of group theoretical properties in terms of these functions. The theory was motivated by the desire to enumerate finite groups of given order, and the analogy with Mikhail Gromov 's notion of word growth .
Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry . Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry.
Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...
A function with quasi-polynomial growth is also said to be quasi-polynomially bounded. [ 1 ] Quasi-polynomial growth has been used in the analysis of algorithms to describe certain algorithms whose computational complexity is not polynomial, but is substantially smaller than exponential .