For groups $(G,+)$ and $(H,\cdot)$ the wreath product $G \wr H$ is the set $W = G^H \times H$ (the set of all functions times $H$, so a typical element is $(f,h)$ where $f : H \to G$ and $h \in H$) together with the composition $$ (f, h)(f', h') = (g, hh') $$ where $g$ is defined, for each $q \in H$ by $$ g(q) = f(q) + f'(qh). $$ (the additive notation does not imply it to be commutative!) Now the lamp lighter group is $\mathbb Z_2 \wr \mathbb Z$. So as I see it with the above mentioned definitions for example $f(n) = n \mbox{mod} 2$ and $f'(n) = 1_{\mathbb N}$ (indicator function of the natural numbers) then $$ (f, 3) \mbox{ and } (f', 5) $$ would be both elements of the lamp lighter group, and their product would be $(g, 15)$ with $g(n) = f(n) + f'(n+3)$. But as I read elsewhere, and according to a famous interpretation of this group, as functions are just those allowed for which $\{ n : f(n) = 1 \}$ is finite. But where does this come from, to quote a book:
... consider the lamplighter group $\mathbb Z_2 \wr \mathbb Z$, the wreath product of $\mathbb Z_2$ with $\mathbb Z$. One can define this group by using as a state space $X$ the configuration space of a doubly infinite sequence of lamps (indexed by the inegers), with each lamp being either "on" or "off", and with at most finitely many of the lamps begin "on", together with the position of a lamplighter, located at one of the lamps; more formally, we have $X := (\mathbb Z)^{\mathbb Z}_0 \times \mathbb Z$, where $(\mathbb Z)^{\mathbb Z}_0$ is the space of compactly supported sequences from $\mathbb Z$ to $\mathbb Z_2$. The lamplighter has the ability to toggle the lamp on and off at his or her current location, and also the ability to move left or right...
On wikipedia there is a similar interpretation. But where does this interpretation and also the requirement that just functions whose roots are a cofinite set comes from? I don't see it in the definition of the wreath product, or have I misunderstood something? Hope someone could clarify..
$\endgroup$ 21 Answer
$\begingroup$You speak of "the" definition of wreath product and mention Wikipedia $-$ but there is more than one type of wreath product: there are restricted and unrestricted wreath products (as you will see defined on the Wikipedia page for wreath product).
$\endgroup$ 6
