What are some elementary examples of elements and operations on the Lamplighter group $L$?
I have the definition above which is the infinite sum of infinitely many copies of the cyclic group $C_2$, and I can tentatively work out what this means, but not confidently.
It looks to me like any given element of $L$ comprises the ordered pair $a,z$ where $a$ is a binary string starting and ending in $1$ and $z$ is any integer but I'm not sure about this as I'm struggling to understand the concept of the wreath product.
I want to categorise its similarities and differences to other algebras containing a binary string and an action of $\Bbb Z$ such as the multiplicative monoid $\Bbb Z[\frac12]^+/\langle2\rangle$ or even the group $\Bbb Q^+/\langle2\rangle$.
If I understand correctly then $a,b\in L$ are examples of two elements:
$a=(10100101_2,7)$
$b=(111_2,2)$
Then I'm guessing but I'd say to combine them, offset each by $z$ digits and $\mathrm{XOR}$ adjacent elements to get:
$a\circ b=(1010010100111,2)$
I'd estimate the probability of this being correct is pretty slim.