Growth Rate (group Theory) - Examples

Examples

• A free group with a finite rank k > 1 has an exponential growth rate.

• A finite group has constant growth – polynomial growth of order 0 – and 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. Milnor proved this using the fact that the word metric on is quasi-isometric to the universal cover of M.

• Zd has a polynomial growth rate of order d.

• The discrete Heisenberg group H₃ has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Bass and 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. It was asked by Milnor in 1968 and was finally answered in the positive by 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 3 finite groups (the spherical ones, corresponding to sphere), 3 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).