---
tag: 4GNPFMT7DZ
kind: example
name: Lamplighter group
msc: [20F65, "20E22"]
instance_of: [W1X984PKJ9, CEAT9BESKK]
sources:
  - url: https://en.wikipedia.org/wiki/Lamplighter_group
same_as:
  wikidata: Q6482453
  wikipedia: Lamplighter group
---

## Example

Let $L = \bigoplus_{i \in \Z} \Z/2\Z$ be the group of finitely supported functions $f \colon \Z \to \Z/2\Z$, with pointwise addition.
Let $\Z$ act on $L$ by shifts: $(n \cdot f)(i) = f(i - n)$.
The *lamplighter group* is the [[semidirect product|0H3WZXCZNF]]
$$\Z/2\Z \wr \Z = L \rtimes \Z,$$
the restricted [[wreath product|W1X984PKJ9]] of $\Z/2\Z$ by $\Z$.
An element is a pair $(f, n)$.
Here $f$ says which lamps on the integer line are lit, finitely many of them, and $n$ is the position of the lamplighter.

It is generated by $t = (0, 1)$, which moves the lamplighter one step, and $a = (\delta_0, 0)$, which switches the lamp at $0$.
So it is a [[finitely generated group|082DWWXMRJ]].
It is an extension of the abelian group $L$ by the abelian group $\Z$, so it is [[solvable|0FXJ7YFMMX]] of derived length $2$.
Solvable groups are amenable, so it is [[amenable|CEAT9BESKK]].

## Remarks

The group has exponential [[growth|4CMRM3975W]].
Consider the $2^n$ words $a^{\varepsilon_1} t \, a^{\varepsilon_2} t \cdots a^{\varepsilon_n} t$ with $\varepsilon_i \in \set{0, 1}$, of length at most $2n$.
They give distinct elements, because the lamps lit along the way, at $n$ consecutive positions, record the sequence $(\varepsilon_1, \ldots, \varepsilon_n)$.
So it is a solvable group that is not virtually nilpotent.
It has the infinite presentation $\langle a, t \mid a^2, \ [a, t^k a t^{-k}] \text{ for } k \in \Z \rangle$, and it is not finitely presented.

## Explanations

### L1 human

Picture an infinite street of lamps, one at each integer, and a lamplighter walking along it.
A state is a finite set of lit lamps and a position of the lamplighter.
The two moves are: walk one step, or switch the lamp where you stand.
Combining moves is the group operation.
This simple group is a standard source of examples and counterexamples in [[geometric group theory|2T3218RXTD]].