Which group is this?

Question

Define $G=\left\langle a,b\ |\ a^2=1, (ab^2)^2=1 \right\rangle $.

This is an infinite group whose Cayley graph is best described as a two-dimensional grid. Is it a well-known group? What is known about its main properties?

Thanks in advance!

Answer 1

You can rewrite the second relation to give the alternative presentation $$\langle a,b \mid a^2=1, a^{-1}b^2a=b^{-2} \rangle$$ which is an HNN extension, so you can derive its properties, such as normal form for group elements, from the known properties of HNN-extensions.

I don't know whether it has a better known `name', but it is the quotient group of the Baumslag-Solitar group ${\rm BS}(2,-2) = \langle x,y \mid x^{-1}y^2x = y^{-2} \rangle$ by its centre $\langle x^2 \rangle$.

It is automatic, but not hyperbolic, because it has the free abelian subgroup $\langle (ab)^2,b^2 \rangle$.

Answer 2

I believe the original poster is correct about this group having a Cayley Graph which can be nicely drawn on a grid. Excuse my MS Paint skills. Continue the diagram to infinity. Since $a$ has order $2$ we don't need an arrow, and $b$ has infinite order. Note that the relations hold. By drawing the Cayley Graph this way I do not think I have accidently introduced any new relations.

If one pretends the border of this graph wraps back around, one can construct various quotients. Doing so in the above picture we obtain a group of order $16$, SmallGroup(16,3) in GAP.