Regarding the lampligther group $\mathbb{Z}_2\wr\mathbb{Z}$, wikipedia (and other) gives the following presentation :
$$\langle a, t \mid a^2, (a t^n a t^{-n})^2, n \in \mathbb{Z} \rangle$$
With the generating set:
- $a$ reflects the act of switching the lamp at the current position
- $t$ moves the position by $+1$
I'm looking at the following word : $w = atatat^{-2}at^2at^{-1}at^{-1}$ (which is basically saying I light on in order $0,1,2$ and then light off in order $0,2,1$). We can follow the state of the lamps as (starting with the initial state $e$)
$$\small\begin{array}{l|ccccccccccccc}
w&e&a&t&a&t&a&t^{-2}&a&t^2&a&t^{-1}&a&t^{-1}\\
\hline
\text{position}&0&0&1&1&2&2&0&0&2&2&1&1&0\\
\text{lights on}&\emptyset&\{0\}&\{0\}&\{0,1\}&\{0,1\}&\{0,1,2\}&\{0,1,2\}&\{1,2\}&\{1,2\}&\{1\}&\{1\}&\emptyset&\emptyset\\
\end{array}$$
It follows that $w=e$ and it should be part of the relations. However I don't see how to build it from $(a t^n a t^{-n})^2, n \in \mathbb{Z}$. For me these relations work for "pairs of lights", but they fail to capture other null actions when working on three lights if the light-on and light-off actions are not done in the same order…
However, the lamplighter group is well studied, and I doubt that there would be such a mistake in its presentation. Can anyone explain where I fail?
Best Answer
The relations say that $a$ commutes with $t^nat^{-n}$ for all $n \in {\mathbb Z}$. So, in the group we have $$\begin{align} w&=atatat^{-2}at^2at^{-1}at^{-1}\\ & = atata^2t^{-2}at^2t^{-1}at^{-1} \\ & =atatt^{-2}at^2t^{-1}at^{-1} \\ & =atat^{-1}atat^{-1} \\ & = ata^2t^{-1}att^{-1} \\ &=att^{-1}att^{-1} \\ & =a^2 \\ &=1. \end{align}$$