---
tag: 4CMRM3975W
kind: concept
name: Growth rate of a group
msc: [20F69, 20F65]
sources:
  - url: https://en.wikipedia.org/wiki/Growth_rate_(group_theory)
same_as:
  wikidata: Q4139973
  wikipedia: Growth rate (group theory)
---

## Summary

The growth function of a [[finitely generated group|082DWWXMRJ]] counts how many elements can be written as words of length at most $n$ in the generators.
Its rate of increase, up to a natural equivalence, does not depend on the generating set, and it is a [[quasi-isometry|6FKYSH729T]] invariant.
Groups split into groups of polynomial growth, of exponential growth, and of intermediate growth.
[[Gromov's theorem|RSBFHHPTWX]] identifies the groups of polynomial growth as the virtually nilpotent groups.

## Explanations

### L1 human

Start at the identity of the group and take $n$ steps, each step a multiplication by a generator or its inverse.
Count how many different elements you can reach.
In $\Z$ with generator $1$, after at most $n$ steps you reach the $2n + 1$ integers from $-n$ to $n$, which grows like a line.
In $\Z^2$ you reach a diamond of about $2n^2$ points, which grows like an area.
In a free group on two generators, the number of reachable elements roughly triples with each step, which is exponential growth.