By G. Lorentz

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Suppose g ∈ F2 is of length r and suppose that u and v are positive elements with v = gug−1 . Let g = a1 · · · ar be a reduced expression for g, where ai ∈ {x, y, x−1 , y −1 } for 1 ≤ i ≤ r. We consider three cases. (i): If ℓ(gu) < ℓ(g) + ℓ(u), then the first letter z of u must be a−1 r ∈ {x, y}. Write u = zu1 and g = hz −1 for some u1 ∈ {x, y}∗ and some h ∈ F2 . Then v = gug−1 = (hz −1 )(zu1 )(hz −1 )−1 = h(u1 z)h−1 . Since ℓ(h) < r and z ∈ {x, y}, it follows from the induction hypothesis that u1 z and v are conjugate in {x, y}∗ .

If ǫ = p1 , p2 , . . , pr denote the sequence of palindromic prefixes of Pal(w) different than Pal(w) listed in order of increasing length, and if z1 , z2 , . . , zr ∈ {x, y} denote the letters in Pal(w) immediately following the prefixes p1 , p2 , . . , pr in Pal(w), then w = z1 z2 · · · zr . 10. If w = vzu, where u does not have an occurrence of the letter z, then Pal(wz) = Pal(w) Pal(v)−1 Pal(w), where the product is evaluated in the free group generated by {x, y}. 11. 7) Let αx = G = (x, xy) and αy = D = (yx, y).

3. The third is explained by our next characterization of Christoffel words. 2 (Mantaci, Restivo, Sciortino [MRS2003]). A word w ∈ {x, y}∗ is a conjugate of a Christoffel word if and only if BW T (w) takes the form y q xp , where p ⊥ q. 1. 1. 1. 2. The Burrows–Wheeler transform BW T is injective on Lyndon words (cf. [CDP2005] for more details): Given a Lyndon word w = a1 a2 · · · an , let b1 b2 · · · bn and c1 c2 · · · cn denote the first and last columns, respectively, of the Burrows–Wheeler matrix.