Lamplighter group


In mathematics, the lamplighter group L of group theory is the restricted wreath product

Introduction

The name of the group comes from viewing the group as acting on a doubly infinite sequence of street lamps each of which may be on or off, and a lamplighter standing at some lamp An equivalent description for this, called the base group of is
an infinite direct sum of copies of the cyclic group where corresponds to a light that is off and 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 gives the position of the lamplighter, and 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, while the generator a means that the state of lamp lk is changed 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 in two ways. The Turing machine has unbounded memory, but has only used a finite amount of memory at any given time. Moreover, the Turing machine's head is analogous to the lamplighter.

Presentation

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. In fact there is no finite presentation for the lamplighter group, that is it is not finitely presented.

Matrix representation

Allowing to be a formal variable, the lamplighter group is isomorphic to the group of matrices
where and ranges over all polynomials in
Using the presentations above, the isomorphism is given by

Generalizations

One can also define lamplighter groups, with, so that "lamps" may have more than just the option of "off" and "on." The classical lamplighter group is recovered when