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# Lamplighter group

In mathematics, the lamplighter group L of group theory is the wreath product Z/2Z ≀ Z. The base group B of L is

\( \bigoplus_{-\infty}^\infty \mathbb{Z}/2\mathbb{Z}, \)

and so L/B is isomorphic to Z.

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

\( \langle a, t \mid a^2, [ t^m a t^{-m} , t^n a t^{-n} ], m, n \in \mathbb{Z} \rangle \) , which may be simplified to

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

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.

The name of the group comes from viewing the group as acting on a doubly infinite sequence of street lamps ..., *l*_{-2}, *l*_{-1}, *l*_{0}, *l*_{1}, *l*_{2}, ..., each of which may be on or off, and a lamplighter standing at some lamp *l*_{k}. 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).

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.

See also

Growth rate (group theory)

References

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

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