On contraction of vertices of the circuits in coset diagrams for $PSL(2,\mathbb{Z})$

Proceedings of the Indian Academy of Sciences. Mathematical sciences

QAISER MUSHTAQ^1¹  ABDUL RAZAQ^2² 

[1] The Islamia University, Bahawalpur, Pakistan^1;University of Education Lahore, Jauharabad Campus, Jauharabad, Pakistan^2

关键词: Modular group;    coset diagrams;    homomorphic images;    projective line over finite field;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Coset diagrams for the action of $PSL(2,\mathbb{Z})$ on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field $F_{q}$ , denoted by $PL(F_{q})$, vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for $PL(F_{q})$ is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by $(xy)^{m_{1}} (xy^{−1)m_{2}}$, that is, $(m_{1},m_{2})$. Let $\alpha$ be the homomorphic image of $(m_{1},m_{2})$ obtained by contracting a pair of vertices $v$, $u$ of $(m_{1},m_{2})$. If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from $\alpha$. In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of $(m_{1},m_{2})$?We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.

【 授权许可】

CC BY