In the mathematical subject of
geometric group theory, the growth rate of a
group with respect to a symmetric
generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n.
Definition
Suppose G is a finitely generated group; and T is a finite symmetric set of
generators
(symmetric means that if then ).
Any element can be expressed as a
word in the T-alphabet
Consider the subset of all elements of G that can be expressed by such a word of length ≤ n
This set is just the
closed ball of radius n in the
word metric d on G with respect to the generating set T:
More geometrically, is the set of vertices in the
Cayley graph with respect to T that are within distance n of the identity.
Given two nondecreasing positive functions a and b one can say that they are equivalent () if there is a constant C such that for all positive integers n,
for example if .
Then the growth rate of the group G can be defined as the corresponding
equivalence class of the function
where denotes the number of elements in the set . Although the function depends on the set of generators T its rate of growth does not (see below) and therefore the rate of growth gives an invariant of a group.
The word metric d and therefore sets depend on the generating set T. However, any two such metrics are
bilipschitz
equivalent in the following sense: for finite symmetric generating sets E, F, there is a positive constant C such that
As an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set.
Polynomial and exponential growth
If
for some we say that G has a polynomial growth rate.
The infimum of such k's is called the order of polynomial growth.
According to
Gromov's theorem, a group of polynomial growth is a
virtually
nilpotent group, i.e. it has a
nilpotent
subgroup of finite
index. In particular, the order of polynomial growth has to be a
natural number and in fact .
If for some we say that G has an
exponential growth rate.
Every
finitely generated G has at most exponential growth, i.e. for some we have .
If grows
more slowly than any exponential function, G has a subexponential growth rate. Any such group is
amenable.
Examples
- A
free group of finite rank has exponential growth rate.
- 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 has exponential growth rate.
John Milnor proved this using the fact that the
word metric on is
quasi-isometric to the
universal cover of M.
- The
free abelian group has a polynomial growth rate of order d.
- The
discrete Heisenberg group 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.
- The
lamplighter group has an exponential growth.
- The existence of groups with intermediate growth, i.e. subexponential but not polynomial was open for many years. The question was asked by Milnor in 1968 and was finally answered in the positive by
Rostislav Grigorchuk in 1984. There are still open questions in this area and a complete picture of which orders of growth are possible and which are not is missing.
- The
triangle groups include infinitely many finite groups (the spherical ones, corresponding to sphere), three groups of quadratic growth (the Euclidean ones, corresponding to Euclidean plane), and infinitely many groups of exponential growth (the hyperbolic ones, corresponding to the hyperbolic plane).
See also
References
Further reading