Lamplighter group

In mathematics, the lamplighter group L of group theory is the restricted wreath product $\mathbf {Z} _{2}\wr \mathbf {Z}$

Introduction

The name of the group comes from viewing the group as acting on a doubly infinite sequence of street lamps $\dots ,-l_{2},-l_{1},l_{1},l_{2},l_{3},\dots$ each of which may be on or off, and a lamplighter standing at some lamp $l_k$ . An equivalent description for this, called the base group $B$ of $L$ is

$B=\bigoplus _{-\infty }^{\infty }\mathbf {Z} _{2}$ ,

an infinite direct sum of copies of the cyclic group ${\mathbf Z}_{2}$ where $0$ corresponds to a light that is off and $1$ corresponds to a light that is on, and the direct sum is used to ensure that only finitely many lights are on at once. An element of ${\mathbf Z}$ gives the position of the lamplighter, and $B$ to encode which bulbs are illuminated.

There are two generators for the group: the generator t increments k, so that the lamplighter moves to the next lamp (t^-1 decrements k), while the generator a means that the state of lamp l_k is changed (from off to on or from on to off.) Group multiplication is done by "following" these operations.

Presentation

The standard presentation for the lamplighter group arises from the wreath product structure

\langle a,t\mid a^{2},[t^{m}at^{{-m}},t^{n}at^{{-n}}],m,n\in {\mathbb {Z}}\rangle

, which may be simplified to

\langle a,t\mid a^{2},(at^{n}at^{-n})^{2},n\in \mathbb {Z} \rangle

.

The generators a and t are intrinsic to the group's notable growth rate, though they are sometimes replaced with a and at, changing the logarithm of the growth rate by at most a factor of 2.

We may assume that only finitely many lamps are lit at any time, since the action of any element of L changes at most finitely many lamps. The number of lamps lit is, however, unbounded. The group action is thus similar to the action of a Turing machine.

Matrix Representation

Allowing $t$ to be a formal variable, the lamplighter group $L$ is isomorphic to the group of matrices

${\begin{pmatrix}t^{k}&p\\0&1\end{pmatrix}}$

where $k\in \mathbf {Z}$ and $p$ ranges over all polynomials in $\mathbf {Z} _{2}[t,t^{-1}]$ .^[1]

Using the presentations above, the isomorphism is given by

$t\mapsto {\begin{pmatrix}t&0\\0&1\end{pmatrix}}$

$a\mapsto {\begin{pmatrix}1&1\\0&1\end{pmatrix}}$ .

Generalizations

One can also define lamplighter groups $L_{n}=\mathbf {Z} _{n}\wr \mathbf {Z}$ , with $n\in {\mathbb N}$ , so that "lamps" may have more than just the option of "off" and "on." The classical lamplighter group is recovered when $n=2.$

References

↑ Clay, Matt; Margalit, Dan, eds. (2017-07-11). Office Hours with a Geometric Group Theorist. Princeton, NJ Oxford: Princeton University Press. ISBN 9780691158662.

Volodymyr Nekrashevych, 2005, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society, ISBN 0-8218-3831-8.

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[1] Clay, Matt; Margalit, Dan, eds. (2017-07-11). Office Hours with a Geometric Group Theorist. Princeton, NJ Oxford: Princeton University Press. ISBN 9780691158662.