TITLE: What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
QUESTION [28 upvotes]: 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!

REPLY [21 votes]: $a^2 = b^2$ says to me "two Mobius bands glued along their boundaries" which then says "the fundamental group of the Klein bottle."  You can hear similar sounds from the presentations $\langle a, b \mid a^p = b^q \rangle$ which give rise to fundamental groups of torus knot complements. (Hmm.  As long as $p, q$ are relatively prime.)