Lamplighter group

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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

$\langle a, t \mid [ t^m a t^{-m} , t^n a t^{-n} ], m, n \in \mathbb{Z} \rangle$ , 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, l₀, l₁, l₂, ... , 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.