I= COSET DIAGRAMS FOR AN INFINITE TRIANGLE GROUP ˆ† (2, 3, 11)
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Title of Thesis
COSET DIAGRAMS FOR AN INFINITE TRIANGLE GROUP ˆ† (2, 3, 11)

Author(s)
Tariq Maqsood
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University, Islamabad
Session
2004
Subject
Mathematics
Number of Pages
128
Keywords (Extracted from title, table of contents and abstract of thesis)
coset diagrams, infinite triangle, conjugacy class, depict homomorphic images, finite index, modular group, triangle groups, parametrization

Abstract
It is known that each conjugacy class of actions of PGL(2, Z) on PL(Fq) = Fq O can be represented by a coset diagram D(o, q), where o E Fq and q is a power of a prime. We have associated each conjugacy class of actions of the infinite triangle group ˆ† (2, 3, 11) = < x, y : x2 = y3 = (x y)11 = 1> on PL(Fq) with a coset diagram D(o, q), where o E Fq. In other words, we have obtained conditions on o and q which guarantee only those coset diagrams which depict homomorphic images of ˆ† (2, 3, 11) in PGL(2, q). We are interested in finding also when the coset diagrams for the actions of PGL(2, Z) on PL(Fq) contain vertices on the vertical line of symmetry. It will enable us to show that for infinitely many values of the prime-power q, the group PGL(2, q) has minimal genus, while also for infinitely many q, the group PSL(2, q) is an H' - group.

We have also answered the following question. Given a fragment of a coset diagram, for what values of q and o can this fragment be found in the coset diagram representing the homomorphic images of ˆ† (2, 3, 11)? The condition for the existence of a fragment in D(o, q) is a polynomial f(z) in Z[z] such that f(o) = O. We have also devised a technique to stitch together small diagrams of homomorphic images of ˆ† (2, 3, 11) to obtain homomorphic images of ˆ† (2, 3, 11) for larger class of numbers not confined to q. By using Cebotarev's Density Theorem, we have also answered the question: with what frequency certain fragments of coset diagrams occur in homomorphic images of ˆ† (2, 3, 11)?

With a subgroup of finite index in ˆ† (2, 3, 11) = < x, y: x2 = y3 = (xy)11= 1 >, we have associated a quintuple of non-negative integers (n, g, e, f, h), with n = q + 1, and q = 11, or q = p, for some prime p with p =± 1 (mod 11), or q = p5, for some prime p =± 2 or ± 3 or ± 4 or ± 5 (mod 11) and n is the number of vertices of a coset diagram, g is genus, e is the number of fixed points of x, f is the number of fixed points of y, and h is the number of fixed points of x y, which satisfies the genus formula 5n = 132(g -1) + 33e + 44 f + 60h. We have shown that all coset diagrams with vertices from PL(Fq) where q = 11, or q = p, for some prime p with p =± 1 (mod 11), or q = p5, for some prime p =± 2 or ± 3 or ± 4 or ± 5 (mod 11) with specification (n, g, e, f, h), satisfying the genus formula 5n = 132 (g - 1) + 33 e + 44 f + 60 h, correspond to a subgroup of finite index n in an infinite triangle group ˆ† (2, 3, 11). This is an important step towards classifying these groups.

The groups Gk,l,m have been studied extensively by H. S. M. Coxeter. We have taken k = 3, l = 11, and answered G. Higman's question for the minimum values of m by using a diagrammatic argument. That is, we have shown that all but finitely many positive integers n, both alternating and symmetric groups occur as homomorphic images of G 3, 11,924.

Finally, by using fragments, we have proved that for a family of positive integers n, all alternating and symmetric groups occur as a homomorphic images of G3,11,m where m = 2r s or m = 2r if r = s for some primes r and s.

Download Full Thesis
10226.83 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
1391.13 KB
2 1 Some Basic Definitions 01
2218.09 KB
  1.1 Modular Group 01
  1.2 Triangle Groups 05
  1.3 Coset Diagrams 16
  1.4 Cebotarev's Density Theorem 23
3 2 Homomorphic Images Of ˆ† (2 ,3,11 ) 25
2529.9 KB
  2.1 Parametrization and Coset Diagrams 25
4 3 Fragments Of Coset Diagrams For ˆ† (2, 3, 11) 58
1813.73 KB
  3.1 Stitching of Coset Diagrams 58
  3.2 Fragments of Coset Diagrams 65
  3.3 Application of Cebotarev's Density Theorem 76
5 4 Subgroups Of Finite Index In ˆ† (2, 3, 11) 81
729.26 KB
  4.1 Genus Formula for ˆ† (2, 3, 11) 81
  4.2 Specification 84
  4.3 Building Blocks for Coset Diagrams 84
  4.4 Subgroups of Finite Index in ˆ† (2, 3, 11) 87
6 5 The Group G3 ,11,924 90
897.33 KB
  5.1 The Group Gk ,l,m 90
  5.2 The Group G3 ,11,m 92
  5.3 Coset Diagrams for G3 11 m and there Composition 93
  5.4 The Quotients of G3, 11, 924 96
7 6 Alternating And Symmetric Groups As Homomorphic Images Of G3, 11, m 101
337.3 KB
  6.1 Actions of G3 ,11,m on PL( Fq ) 101
  6.2 An and Sn as Homomorphic Images of G3 ,11,m 102
8 7 References 106
468.29 KB