TITLE: Is the free group on two generators generated by two elements as a monoid?
QUESTION [10 upvotes]: I was musing about this today and couldn't come up with an answer. Obviously it can be generated as a monoid by the four elements $a$, $b$, $a^{-1}$, and $b^{-1}$.  After some playing around I was able to come up with three elements that generate it as a monoid: $ab$, $ab^{-1}$, and $a^{-1}$.
But I haven't been able to come up with two generators, nor an argument as to why that should be impossible.

REPLY [6 votes]: And, further generalizing the rank 2 case, the $(n+1)$-tuple $(a_1^{-1}, a_2^{-1},\ldots,a_n^{-1},a_1a_2\cdots a_n)$ generates all of the rank $n$ free group as a monoid.