  We consider sets of factors that ca be avoided in square-free words on
two-generator free groups. The elements of the group are presented in terms of
$\{0,1,2,3\}$ such that $0$ and $2$ (resp.,$1$ and $3$) are inverses of each
other so that in a reduced word $02$, $20$, $13$ and $31$ do not occur. A Dean
word is a reduced word that does not contain occurrences of $ww$ for any
nonempty~$w$. Dean showed in 1965 that there exist infinite square-free reduced
words. We show that if $w$ is a Dean word of length at least 59 then there are
at most six reduced words of length 3 avoided by $w$. The six avoided words
belong to four specific sets that are obtained by a permutation of the letters
from each other. Finally, we prove that there exists an infinite Dean word for
each of the four sets of six elements.
