2021/04/14 by Golnaz Badkobeh, Tero Harju, Badkobeh, Golnaz +5
Mathematics · #68R15 #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #acm:68R15 #math.CO #msc:68R15
paper · pdf · doi:10.48550/arxiv.2104.06837
19 pages Lemma 25 added
arxiv created 2021/08/24 · arxiv updated 2021/08/25
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.