Abelian squares are avoidable over four-letter alphabet as shown by V. Keränen in

V. Keränen, Abelian squares are avoidable on 4 letters, Proc. ICALP '92, Lecture Notes in Comp. Sci. 623, Springer, Berlin (1992), pp. 41–52

Keränen's infinite word is given as the fixed point of a 85-uniform morphism g85 defined by:

g85(a)=abcacdcbcdcadcdbdabacabadbabcbdbcbacbcdcacbabdabacadcbcdcacdbcbacbcd
          cacdcbdcdadbdcbca

g85(b)=σ(h(a))

g85(c)=σ2(h(a))

g85(d)=σ3(h(a))

where σ is a cyclic permutation of letters:

σ: a ↦ b, b ↦ c, c ↦ d, d ↦ a

-- StepanHolub - 10 Mar 2012

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Topic revision: r2 - 2012-03-10 - StepanHolub
 
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