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Les nombres de Catalan et le groupe modulaire PSL2(Z)

Abstract : In this thesis, we study a morphism of mono"id mu between the free mono"id on the alphabet of integers nband the modular group PSL2(zb) considered as a mono"id, defined for all integer aby mu(a)=begin{pmatrix} 0 & -1 1 & a+1 end{pmatrix}. The Catalan Numbers arised naturally in the study ofsubsets of the kernel of the morphism mu.Firstly, we introduce two rewriting systems, one on the finite alphabet {0,1}, and the other on the infinite alphabet of integers nb. We proove that bothof these rewriting systems defines a mono"id presentation of PSL2(zb) by generators and relations.On another note, we introduce the morphism of loop associated to the abelianised of the universal covering group of PSL2(zb), the group B3 ofbraid group on 3 strands. In two different contexts, the morphism of loop is associated to the number of "half-turns".Then, in the fourth and the fifth parts, we numerate subsets of the kernel of mu{|{0,1}} and of the kernel of mu,bi-graduated by the morphism of lengthand the morphism of loop. The sequences of Catalan numbers and other diagonals of the Catalan triangle come into the results.Lastly, we present the geometrical origin of this research : we detail the connection between our first aim,which was the study of convex integer polygones ofminimal area, and our interest for the mono"id generated by these particular matrices of PSL2(zb).
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Christelle Guichard. Les nombres de Catalan et le groupe modulaire PSL2(Z). Théorie des nombres [math.NT]. Université Grenoble Alpes, 2018. Français. ⟨NNT : 2018GREAM057⟩. ⟨tel-02024805⟩

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