  We consider sets of factors that can 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
02, 20, 13 and 31 do not occur in a reduced word. A Dean word is a reduced word
that does not contain occurrences of $uu$ for any nonempty $u$. 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 $w$. We construct an infinite Dean word avoiding six
reduced words of length~3. We also construct infinite Dean words with low
critical exponent and avoiding fewer reduced words of length 3. Finally, we
show that the minimal frequency of a letter in a Dean word is $8/59$ and the
growth rate is close to 1.45818.
