1. Vladimir DREMOV, Moscow State University; George SHABAT, Moscow State University and Institute of Theoretic and Experimental Physics. On the combinatorial description of Fried families Riemann Surfaces and Dessins d’Enfants, a conference on the occasion of Jurgen Wolfart’s 65 th Birthday. Castro Urdiales, 2010, May 24-28

2. Plan 0. Belyi theorem. Curves over number fields and over complex numbers. 1. Belyi pairs and Fried families. 2. Drawing Fried families. 3. Category equivalencies 4. Examples and final comments.

3. 1. Belyi pairs and Fried families Arbitrary ground field (algebraically closed, the characteristic does not divide degrees). A curve X of genus g, a function f on it of degree d. Belyi case: #CritVal(f) = 3. g+d+1 “unknowns”, g+d+1 “equations”. Fried case: #CritVal(f) = 4. g+2d+3 “unknowns”, g+2d+2 “equations”.

4. Belyi sphere

5. Colored dessin

6. Fried sphere

7. Preimage of the Fried sphere

8. 3. CATEGORIES 0. Complex Fried families. 1. “Square” sets. 2. Special quasifuchsian families Details on the blackboard.

9. 4. EXAMPLES. Fried polynomials of degree 6., where * is one of (4,1,1); (3,2,1); (2,2,2). Paths connecting the trees in the Betrema-Pere-Zvonkin forest!

10. Fried family of polynomials of degree 5

11. Genus 1 and degree 2 Fried family (with trivial base)

12. <d|d|d|d><d|d|d|d> Genus = d -1 The function x is the Fried one on the affine plane curve defined by the equation where gcd(k,d)=gcd(l,d)=gcd(m,d)=gcd(k+l+m,d)=1. Each of {0,1, t, ∞} has the only preimage on the normal projective model of X

13. FINAL COMMENTS 1. Groups of Belyi pairs (including Galois orbits) often lie on the Fried curves. 2. Some computational phenomena cry out for theoretical explanation. 2.1. Primes in the denominators (bad reduction, arithmetic of the moduli spaces?). 2.2. Enormous multiplicities of the parasitic solutions (intersection theory on the moduli spaces?). 2.3. …