Editing Growth rate (group theory) (section)

==Examples==

* A [[free group]] of finite rank <math>k > 1</math> has exponential growth rate.
* A [[finite group]] has constant growth—that is, polynomial growth of order 0—and this includes [[fundamental group]]s of [[manifold]]s whose [[universal cover]] is [[compact space|compact]].
* If ''M'' is a [[closed manifold|closed]] [[Curvature of Riemannian manifolds|negatively curved]] [[Riemannian manifold]] then its [[fundamental group]] <math>\pi_1(M)</math> has exponential growth rate. [[John Milnor]] proved this using the fact that the [[word metric]] on <math>\pi_1(M)</math> is [[Glossary of Riemannian and metric geometry#Q|quasi-isometric]] to the [[Covering map|universal cover]] of ''M''.
* The [[free abelian group]] <math>\Z^d</math> has a polynomial growth rate of order ''d''.
* The [[discrete Heisenberg group]] <math>H_3</math> 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 on groups of polynomial growth|Gromov's theorem]].
* The [[lamplighter group]] has an exponential growth.  <!-- This is a rare example of a solvable group with 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 group]]s 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).