In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}\*\mathbb{Z}/2$ because clearly every element can be written as a product of $b$'s (only to the power 1) and powers of $a$. But this turns out to be far from clear (and likely wrong). I assume this must be a well-known group to group theorists, so I'm curious if it's isomorphic to something that can be described by other means (or what's known about it in general). 


Thanks!



What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
Setting $c:=b^{-1}$ one obtains the presentation
$$G= \langle a,c \, | \, a^2c^2=1 \rangle,$$
which is the fundamental group of the Klein bottle.


It is well known that another presentation of such a group is 
$$G= \langle x,y \,|\, x^{-1}yx=y^{-1} \rangle,$$
and this allows one to write $G$ as a semi-direct product of infinite cyclic groups, namely $$G= \mathbb{Z} \rtimes\_{\sigma} \mathbb{Z}$$
where $\sigma \colon \mathbb{Z} \to \mathbb{Z}$ is defined by $\sigma(y)=y^{-1}$ and $\mathbb{Z}$ is written multiplicatively.


