Lamplighter group

In mathematics, the lamplighter group L of group theory is the restricted wreath product Z 2 ≀ Z {displaystyle mathbf {Z} _{2}wr mathbf {Z} } In mathematics, the lamplighter group L of group theory is the restricted wreath product Z 2 ≀ Z {displaystyle mathbf {Z} _{2}wr mathbf {Z} } 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 , l 3 , … {displaystyle dots ,l_{-2},l_{-1},l_{0},l_{1},l_{2},l_{3},dots } each of which may be on or off, and a lamplighter standing at some lamp l k {displaystyle l_{k}} . An equivalent description for this, called the base group B {displaystyle B} of L {displaystyle L} is B = ⨁ − ∞ ∞ Z 2 {displaystyle B=igoplus _{-infty }^{infty }mathbf {Z} _{2}} , an infinite direct sum of copies of the cyclic group Z 2 {displaystyle mathbf {Z} _{2}} where 0 {displaystyle 0} corresponds to a light that is off and 1 {displaystyle 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 Z {displaystyle mathbf {Z} } gives the position of the lamplighter, and B {displaystyle 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 lk is changed (from off to on or from on to off.) Group multiplication is done by 'following' these operations. 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. The standard presentation for the lamplighter group arises from the wreath product structure 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. This presentation is not finite (it has infinitely many relations). In fact there is no finite presentation for the lamplighter group, that is it is not finitely presented.